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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
(x-3)/(x^2-8)
-
x^4-2*x^3+1
-
x^4-2*x^3+5
-
x^4/4-2*x^2+1
-
Expresiones idénticas
- dos /(x+ nueve)^ dos - nueve *cos(tres *x)- dos
- 2 dividir por (x más 9) al cuadrado menos 9 multiplicar por coseno de (3 multiplicar por x) menos 2
- dos dividir por (x más nueve) en el grado dos menos nueve multiplicar por coseno de (tres multiplicar por x) menos dos
- 2/(x+9)2-9*cos(3*x)-2
- 2/x+92-9*cos3*x-2
- 2/(x+9)²-9*cos(3*x)-2
- 2/(x+9) en el grado 2-9*cos(3*x)-2
- 2/(x+9)^2-9cos(3x)-2
- 2/(x+9)2-9cos(3x)-2
- 2/x+92-9cos3x-2
- 2/x+9^2-9cos3x-2
- 2 dividir por (x+9)^2-9*cos(3*x)-2
-
Expresiones semejantes
- 2/(x-9)^2-9*cos(3*x)-2
- 2/(x+9)^2+9*cos(3*x)-2
- 2/(x+9)^2-9*cos(3*x)+2
-
Expresiones con funciones
- Coseno cos
- cos(1)/((3*x))
- cos(x)-sin(2)*5*x+e/x
- cos(arccotg√3x^5)
- cos(-2*pi+6*x)
- cos(x)-(1+2x^2)^(1/4)
Gráfico de la función y = 2/(x+9)^2-9*cos(3*x)-2
Solución
Ha introducido
[src]
2
f(x) = -------- - 9*cos(3*x) - 2
2
(x + 9)
$$f{\left(x \right)} = \left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2$$
f = -9*cos(3*x) + 2/(x + 9)^2 - 2
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = -9$$
$$x_{1} = -9$$
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
$$\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2 = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 44.5805671599382$$
$$x_{2} = 16.1569843867817$$
$$x_{3} = -23.6362879910384$$
$$x_{4} = 78.0909052450737$$
$$x_{5} = 49.6672080578059$$
$$x_{6} = 45.4784213778843$$
$$x_{7} = -42.4861307686988$$
$$x_{8} = 11.9682469292956$$
$$x_{9} = -80.1852953712487$$
$$x_{10} = 71.80771831968$$
$$x_{11} = 22.4401265102882$$
$$x_{12} = -85.2719157847567$$
$$x_{13} = 66.4223601590811$$
$$x_{14} = 95.7438851618192$$
$$x_{15} = -5.69181491956718$$
$$x_{16} = 32.0141778453195$$
$$x_{17} = -47.5728419433885$$
$$x_{18} = -233.076151325562$$
$$x_{19} = -75.9965032330992$$
$$x_{20} = 38.2973743548123$$
$$x_{21} = 86.4684873354534$$
$$x_{22} = 48.7693610632215$$
$$x_{23} = -69.713314241322$$
$$x_{24} = -53.8560139478374$$
$$x_{25} = 34.1085772295306$$
$$x_{26} = 0.597471902690822$$
$$x_{27} = -62.2335834082194$$
$$x_{28} = 97.8382799953761$$
$$x_{29} = -45.4784528725154$$
$$x_{30} = -58.0447979783889$$
$$x_{31} = 14.0626120756148$$
$$x_{32} = -60.1391905467204$$
$$x_{33} = -9.95544662407538$$
$$x_{34} = -73.9021070206743$$
$$x_{35} = 24.5345123125316$$
$$x_{36} = 20.3457427694818$$
$$x_{37} = 164.858919129473$$
$$x_{38} = -86.4684830118034$$
$$x_{39} = 53.8559954186518$$
$$x_{40} = -71.8077106953627$$
$$x_{41} = -34.1084976066077$$
$$x_{42} = 81.0831218824425$$
$$x_{43} = -51.7616226346767$$
$$x_{44} = -99.9326776298639$$
$$x_{45} = -97.8382829657979$$
$$x_{46} = 88.5628827918914$$
$$x_{47} = -7.83441958465795$$
$$x_{48} = -95.7438883338631$$
$$x_{49} = -89.4607046651693$$
$$x_{50} = -11.9766247111707$$
$$x_{51} = -21.5417645776835$$
$$x_{52} = -55.9504057566857$$
$$x_{53} = 18.2513617475986$$
$$x_{54} = -16.1583462479383$$
$$x_{55} = 51.7616016647378$$
$$x_{56} = 27.8253767818785$$
$$x_{57} = -93.6494937372787$$
$$x_{58} = 58.0447832956967$$
$$x_{59} = -65.5245208686328$$
$$x_{60} = 82.2796963480715$$
$$x_{61} = 29.9197777514219$$
$$x_{62} = -32.0140795740169$$
$$x_{63} = -36.2029105499022$$
$$x_{64} = -29.9196543149725$$
$$x_{65} = -1.4974477358751$$
$$x_{66} = 80.1853008123896$$
$$x_{67} = -27.8252184403014$$
$$x_{68} = 36.2029760324822$$
$$x_{69} = 84.3740918549255$$
$$x_{70} = 9.87389229863684$$
$$x_{71} = -82.2796913184005$$
$$x_{72} = 73.9021140025122$$
$$x_{73} = -78.0908993461156$$
$$x_{74} = -91.555099179677$$
$$x_{75} = 99.9326748442898$$
$$x_{76} = -3.59309020692213$$
$$x_{77} = 93.649490344885$$
$$x_{78} = -14.0654272421439$$
$$x_{79} = 69.7133225899015$$
$$x_{80} = -49.6672319219022$$
$$x_{81} = -67.6189176398301$$
$$x_{82} = -18.2521467008472$$
$$x_{83} = 42.4861698607794$$
$$x_{84} = -38.2973198056781$$
$$x_{85} = 60.1391773896165$$
$$x_{86} = 62.2335715711739$$
$$x_{87} = -25.7307663132342$$
$$x_{88} = 40.3917722762852$$
$$x_{89} = 64.3279658305012$$
$$x_{90} = 75.9965096429679$$
$$x_{91} = 55.9503893008989$$
$$x_{92} = 879.047646628447$$
$$x_{93} = 5.6852410825241$$
$$x_{94} = 7.77955374934343$$
o sea hay que resolver la ecuación:
$$\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2 = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 44.5805671599382$$
$$x_{2} = 16.1569843867817$$
$$x_{3} = -23.6362879910384$$
$$x_{4} = 78.0909052450737$$
$$x_{5} = 49.6672080578059$$
$$x_{6} = 45.4784213778843$$
$$x_{7} = -42.4861307686988$$
$$x_{8} = 11.9682469292956$$
$$x_{9} = -80.1852953712487$$
$$x_{10} = 71.80771831968$$
$$x_{11} = 22.4401265102882$$
$$x_{12} = -85.2719157847567$$
$$x_{13} = 66.4223601590811$$
$$x_{14} = 95.7438851618192$$
$$x_{15} = -5.69181491956718$$
$$x_{16} = 32.0141778453195$$
$$x_{17} = -47.5728419433885$$
$$x_{18} = -233.076151325562$$
$$x_{19} = -75.9965032330992$$
$$x_{20} = 38.2973743548123$$
$$x_{21} = 86.4684873354534$$
$$x_{22} = 48.7693610632215$$
$$x_{23} = -69.713314241322$$
$$x_{24} = -53.8560139478374$$
$$x_{25} = 34.1085772295306$$
$$x_{26} = 0.597471902690822$$
$$x_{27} = -62.2335834082194$$
$$x_{28} = 97.8382799953761$$
$$x_{29} = -45.4784528725154$$
$$x_{30} = -58.0447979783889$$
$$x_{31} = 14.0626120756148$$
$$x_{32} = -60.1391905467204$$
$$x_{33} = -9.95544662407538$$
$$x_{34} = -73.9021070206743$$
$$x_{35} = 24.5345123125316$$
$$x_{36} = 20.3457427694818$$
$$x_{37} = 164.858919129473$$
$$x_{38} = -86.4684830118034$$
$$x_{39} = 53.8559954186518$$
$$x_{40} = -71.8077106953627$$
$$x_{41} = -34.1084976066077$$
$$x_{42} = 81.0831218824425$$
$$x_{43} = -51.7616226346767$$
$$x_{44} = -99.9326776298639$$
$$x_{45} = -97.8382829657979$$
$$x_{46} = 88.5628827918914$$
$$x_{47} = -7.83441958465795$$
$$x_{48} = -95.7438883338631$$
$$x_{49} = -89.4607046651693$$
$$x_{50} = -11.9766247111707$$
$$x_{51} = -21.5417645776835$$
$$x_{52} = -55.9504057566857$$
$$x_{53} = 18.2513617475986$$
$$x_{54} = -16.1583462479383$$
$$x_{55} = 51.7616016647378$$
$$x_{56} = 27.8253767818785$$
$$x_{57} = -93.6494937372787$$
$$x_{58} = 58.0447832956967$$
$$x_{59} = -65.5245208686328$$
$$x_{60} = 82.2796963480715$$
$$x_{61} = 29.9197777514219$$
$$x_{62} = -32.0140795740169$$
$$x_{63} = -36.2029105499022$$
$$x_{64} = -29.9196543149725$$
$$x_{65} = -1.4974477358751$$
$$x_{66} = 80.1853008123896$$
$$x_{67} = -27.8252184403014$$
$$x_{68} = 36.2029760324822$$
$$x_{69} = 84.3740918549255$$
$$x_{70} = 9.87389229863684$$
$$x_{71} = -82.2796913184005$$
$$x_{72} = 73.9021140025122$$
$$x_{73} = -78.0908993461156$$
$$x_{74} = -91.555099179677$$
$$x_{75} = 99.9326748442898$$
$$x_{76} = -3.59309020692213$$
$$x_{77} = 93.649490344885$$
$$x_{78} = -14.0654272421439$$
$$x_{79} = 69.7133225899015$$
$$x_{80} = -49.6672319219022$$
$$x_{81} = -67.6189176398301$$
$$x_{82} = -18.2521467008472$$
$$x_{83} = 42.4861698607794$$
$$x_{84} = -38.2973198056781$$
$$x_{85} = 60.1391773896165$$
$$x_{86} = 62.2335715711739$$
$$x_{87} = -25.7307663132342$$
$$x_{88} = 40.3917722762852$$
$$x_{89} = 64.3279658305012$$
$$x_{90} = 75.9965096429679$$
$$x_{91} = 55.9503893008989$$
$$x_{92} = 879.047646628447$$
$$x_{93} = 5.6852410825241$$
$$x_{94} = 7.77955374934343$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\frac{2 \left(- 2 x - 18\right)}{\left(x + 9\right)^{4}} + 27 \sin{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -92.1533845911891$$
$$x_{2} = 50.2654826946666$$
$$x_{3} = 28.274332928754$$
$$x_{4} = 15.7079599940573$$
$$x_{5} = -13.6130651210836$$
$$x_{6} = 54.4542728555055$$
$$x_{7} = 17.802355805527$$
$$x_{8} = -94.2477796874062$$
$$x_{9} = -77.4926189422375$$
$$x_{10} = 78.5398162661312$$
$$x_{11} = -81.6814091219536$$
$$x_{12} = 76.4454211581911$$
$$x_{13} = 98.4365698523018$$
$$x_{14} = 32.4631233943248$$
$$x_{15} = -21.9911260516957$$
$$x_{16} = -11.516072706381$$
$$x_{17} = -63.8790503242099$$
$$x_{18} = 74.3510260496794$$
$$x_{19} = 94.2477796525614$$
$$x_{20} = -57.5958648855072$$
$$x_{21} = 90.0589894537109$$
$$x_{22} = 48.1710876193121$$
$$x_{23} = -20.9439800058931$$
$$x_{24} = 19.8967514261592$$
$$x_{25} = -59.6902600390645$$
$$x_{26} = 52.3598777735875$$
$$x_{27} = 61.7846553813613$$
$$x_{28} = -61.7846551848215$$
$$x_{29} = 8.37758981993514$$
$$x_{30} = -46.0766932215348$$
$$x_{31} = 43.9822974822911$$
$$x_{32} = 72.2566309405206$$
$$x_{33} = 6.2831991406822$$
$$x_{34} = 2.09443126500936$$
$$x_{35} = 87.9645943546813$$
$$x_{36} = -85.8701993068389$$
$$x_{37} = -4.18834691045651$$
$$x_{38} = -33.5103249920186$$
$$x_{39} = 81.6814090595594$$
$$x_{40} = -70.1622357143355$$
$$x_{41} = -19.8967153057235$$
$$x_{42} = -35.6047193630843$$
$$x_{43} = 41.887902422605$$
$$x_{44} = -54.4542731880597$$
$$x_{45} = 46.0766925482278$$
$$x_{46} = -72.2566308374655$$
$$x_{47} = 92.1533845530133$$
$$x_{48} = -15.707799648569$$
$$x_{49} = -2.09424515410351$$
$$x_{50} = -99.4837672970172$$
$$x_{51} = -55.501469722314$$
$$x_{52} = -24.08552929307$$
$$x_{53} = -90.058989495627$$
$$x_{54} = 65.9734456082059$$
$$x_{55} = 4.18881173052025$$
$$x_{56} = -30.3687239236679$$
$$x_{57} = -79.5870140313522$$
$$x_{58} = -65.9734454583572$$
$$x_{59} = -43.9822983037899$$
$$x_{60} = 63.8790504954171$$
$$x_{61} = 10.4719822007085$$
$$x_{62} = 83.775804157568$$
$$x_{63} = 26.1799376457146$$
$$x_{64} = 21.9911469160688$$
$$x_{65} = -68.0678405881596$$
$$x_{66} = -41.8879034361098$$
$$x_{67} = -98.4365698815089$$
$$x_{68} = 68.0678407198953$$
$$x_{69} = 70.1622358306267$$
$$x_{70} = 85.8701992559553$$
$$x_{71} = 100.530964952454$$
$$x_{72} = 34.5575185919221$$
$$x_{73} = 37.6991123279741$$
$$x_{74} = 80.634211373565$$
$$x_{75} = 59.6902602658396$$
$$x_{76} = 96.3421747523313$$
$$x_{77} = -37.699113932227$$
$$x_{78} = -17.8022859620081$$
$$x_{79} = -28.2743269856721$$
$$x_{80} = -6.28072934545638$$
$$x_{81} = 56.5486679399572$$
$$x_{82} = -83.7758042138392$$
$$x_{83} = -48.1710881766769$$
$$x_{84} = -29.3215373179469$$
$$x_{85} = 39.7935073705685$$
$$x_{86} = 6.77388221724442 \cdot 10^{-5}$$
$$x_{87} = -87.9645944008088$$
$$x_{88} = 24.0855423140059$$
$$x_{89} = -39.7935086366804$$
$$x_{90} = 30.3687281753803$$
$$x_{91} = -26.1799290409932$$
$$x_{92} = -50.2654831602082$$
Signos de extremos en los puntos:
Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
$$x_{1} = -92.1533845911891$$
$$x_{2} = 50.2654826946666$$
$$x_{3} = 54.4542728555055$$
$$x_{4} = -94.2477796874062$$
$$x_{5} = -77.4926189422375$$
$$x_{6} = -81.6814091219536$$
$$x_{7} = 98.4365698523018$$
$$x_{8} = 94.2477796525614$$
$$x_{9} = 90.0589894537109$$
$$x_{10} = 48.1710876193121$$
$$x_{11} = -20.9439800058931$$
$$x_{12} = 52.3598777735875$$
$$x_{13} = 8.37758981993514$$
$$x_{14} = -46.0766932215348$$
$$x_{15} = 43.9822974822911$$
$$x_{16} = 6.2831991406822$$
$$x_{17} = 2.09443126500936$$
$$x_{18} = 87.9645943546813$$
$$x_{19} = -85.8701993068389$$
$$x_{20} = -4.18834691045651$$
$$x_{21} = -33.5103249920186$$
$$x_{22} = 81.6814090595594$$
$$x_{23} = -35.6047193630843$$
$$x_{24} = 41.887902422605$$
$$x_{25} = -54.4542731880597$$
$$x_{26} = 46.0766925482278$$
$$x_{27} = 92.1533845530133$$
$$x_{28} = -2.09424515410351$$
$$x_{29} = -90.058989495627$$
$$x_{30} = 4.18881173052025$$
$$x_{31} = -79.5870140313522$$
$$x_{32} = -43.9822983037899$$
$$x_{33} = 10.4719822007085$$
$$x_{34} = 83.775804157568$$
$$x_{35} = -41.8879034361098$$
$$x_{36} = -98.4365698815089$$
$$x_{37} = 85.8701992559553$$
$$x_{38} = 100.530964952454$$
$$x_{39} = 37.6991123279741$$
$$x_{40} = 96.3421747523313$$
$$x_{41} = -37.699113932227$$
$$x_{42} = -6.28072934545638$$
$$x_{43} = 56.5486679399572$$
$$x_{44} = -83.7758042138392$$
$$x_{45} = -48.1710881766769$$
$$x_{46} = -29.3215373179469$$
$$x_{47} = 39.7935073705685$$
$$x_{48} = 6.77388221724442 \cdot 10^{-5}$$
$$x_{49} = -87.9645944008088$$
$$x_{50} = -39.7935086366804$$
$$x_{51} = -50.2654831602082$$
Puntos máximos de la función:
$$x_{51} = 28.274332928754$$
$$x_{51} = 15.7079599940573$$
$$x_{51} = -13.6130651210836$$
$$x_{51} = 17.802355805527$$
$$x_{51} = 78.5398162661312$$
$$x_{51} = 76.4454211581911$$
$$x_{51} = 32.4631233943248$$
$$x_{51} = -21.9911260516957$$
$$x_{51} = -11.516072706381$$
$$x_{51} = -63.8790503242099$$
$$x_{51} = 74.3510260496794$$
$$x_{51} = -57.5958648855072$$
$$x_{51} = 19.8967514261592$$
$$x_{51} = -59.6902600390645$$
$$x_{51} = 61.7846553813613$$
$$x_{51} = -61.7846551848215$$
$$x_{51} = 72.2566309405206$$
$$x_{51} = -70.1622357143355$$
$$x_{51} = -19.8967153057235$$
$$x_{51} = -72.2566308374655$$
$$x_{51} = -15.707799648569$$
$$x_{51} = -99.4837672970172$$
$$x_{51} = -55.501469722314$$
$$x_{51} = -24.08552929307$$
$$x_{51} = 65.9734456082059$$
$$x_{51} = -30.3687239236679$$
$$x_{51} = -65.9734454583572$$
$$x_{51} = 63.8790504954171$$
$$x_{51} = 26.1799376457146$$
$$x_{51} = 21.9911469160688$$
$$x_{51} = -68.0678405881596$$
$$x_{51} = 68.0678407198953$$
$$x_{51} = 70.1622358306267$$
$$x_{51} = 34.5575185919221$$
$$x_{51} = 80.634211373565$$
$$x_{51} = 59.6902602658396$$
$$x_{51} = -17.8022859620081$$
$$x_{51} = -28.2743269856721$$
$$x_{51} = 24.0855423140059$$
$$x_{51} = 30.3687281753803$$
$$x_{51} = -26.1799290409932$$
Decrece en los intervalos
$$\left[100.530964952454, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.4365698815089\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\frac{2 \left(- 2 x - 18\right)}{\left(x + 9\right)^{4}} + 27 \sin{\left(3 x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -92.1533845911891$$
$$x_{2} = 50.2654826946666$$
$$x_{3} = 28.274332928754$$
$$x_{4} = 15.7079599940573$$
$$x_{5} = -13.6130651210836$$
$$x_{6} = 54.4542728555055$$
$$x_{7} = 17.802355805527$$
$$x_{8} = -94.2477796874062$$
$$x_{9} = -77.4926189422375$$
$$x_{10} = 78.5398162661312$$
$$x_{11} = -81.6814091219536$$
$$x_{12} = 76.4454211581911$$
$$x_{13} = 98.4365698523018$$
$$x_{14} = 32.4631233943248$$
$$x_{15} = -21.9911260516957$$
$$x_{16} = -11.516072706381$$
$$x_{17} = -63.8790503242099$$
$$x_{18} = 74.3510260496794$$
$$x_{19} = 94.2477796525614$$
$$x_{20} = -57.5958648855072$$
$$x_{21} = 90.0589894537109$$
$$x_{22} = 48.1710876193121$$
$$x_{23} = -20.9439800058931$$
$$x_{24} = 19.8967514261592$$
$$x_{25} = -59.6902600390645$$
$$x_{26} = 52.3598777735875$$
$$x_{27} = 61.7846553813613$$
$$x_{28} = -61.7846551848215$$
$$x_{29} = 8.37758981993514$$
$$x_{30} = -46.0766932215348$$
$$x_{31} = 43.9822974822911$$
$$x_{32} = 72.2566309405206$$
$$x_{33} = 6.2831991406822$$
$$x_{34} = 2.09443126500936$$
$$x_{35} = 87.9645943546813$$
$$x_{36} = -85.8701993068389$$
$$x_{37} = -4.18834691045651$$
$$x_{38} = -33.5103249920186$$
$$x_{39} = 81.6814090595594$$
$$x_{40} = -70.1622357143355$$
$$x_{41} = -19.8967153057235$$
$$x_{42} = -35.6047193630843$$
$$x_{43} = 41.887902422605$$
$$x_{44} = -54.4542731880597$$
$$x_{45} = 46.0766925482278$$
$$x_{46} = -72.2566308374655$$
$$x_{47} = 92.1533845530133$$
$$x_{48} = -15.707799648569$$
$$x_{49} = -2.09424515410351$$
$$x_{50} = -99.4837672970172$$
$$x_{51} = -55.501469722314$$
$$x_{52} = -24.08552929307$$
$$x_{53} = -90.058989495627$$
$$x_{54} = 65.9734456082059$$
$$x_{55} = 4.18881173052025$$
$$x_{56} = -30.3687239236679$$
$$x_{57} = -79.5870140313522$$
$$x_{58} = -65.9734454583572$$
$$x_{59} = -43.9822983037899$$
$$x_{60} = 63.8790504954171$$
$$x_{61} = 10.4719822007085$$
$$x_{62} = 83.775804157568$$
$$x_{63} = 26.1799376457146$$
$$x_{64} = 21.9911469160688$$
$$x_{65} = -68.0678405881596$$
$$x_{66} = -41.8879034361098$$
$$x_{67} = -98.4365698815089$$
$$x_{68} = 68.0678407198953$$
$$x_{69} = 70.1622358306267$$
$$x_{70} = 85.8701992559553$$
$$x_{71} = 100.530964952454$$
$$x_{72} = 34.5575185919221$$
$$x_{73} = 37.6991123279741$$
$$x_{74} = 80.634211373565$$
$$x_{75} = 59.6902602658396$$
$$x_{76} = 96.3421747523313$$
$$x_{77} = -37.699113932227$$
$$x_{78} = -17.8022859620081$$
$$x_{79} = -28.2743269856721$$
$$x_{80} = -6.28072934545638$$
$$x_{81} = 56.5486679399572$$
$$x_{82} = -83.7758042138392$$
$$x_{83} = -48.1710881766769$$
$$x_{84} = -29.3215373179469$$
$$x_{85} = 39.7935073705685$$
$$x_{86} = 6.77388221724442 \cdot 10^{-5}$$
$$x_{87} = -87.9645944008088$$
$$x_{88} = 24.0855423140059$$
$$x_{89} = -39.7935086366804$$
$$x_{90} = 30.3687281753803$$
$$x_{91} = -26.1799290409932$$
$$x_{92} = -50.2654831602082$$
Signos de extremos en los puntos:
(-92.15338459118914, -10.9997107521536)
(50.26548269466663, -10.9994305883538)
(28.27433292875405, 7.00143949521193)
(15.707959994057303, 7.00327609252761)
(-13.613065121083636, 7.09397308045595)
(54.4542728555055, -10.9995032838811)
(17.802355805527025, 7.00278409479037)
(-94.24777968740617, -10.9997247902326)
(-77.49261894223754, -10.9995736733795)
(78.53981626613123, 7.00026098691498)
(-81.68140912195358, -10.999621397638)
(76.44542115819112, 7.0002739380789)
(98.43656985230177, -10.9998267290615)
(32.46312339432476, 7.00116333812243)
(-21.991126051695726, 7.01185047202177)
(-11.516072706380985, 7.31553543773142)
(-63.87905032420993, 7.00066407452241)
(74.35102604967943, 7.00028787774658)
(94.24777965256139, -10.9998123845696)
(-57.59586488550725, 7.00084689849293)
(90.05898945371086, -10.9997961821531)
(48.171087619312125, -10.9993881047398)
(-20.9439800058931, -10.9859804874399)
(19.89675142615922, 7.00239514560858)
(-59.69026003906449, 7.00077836079707)
(52.35987777358746, -10.9994687963806)
(61.78465538136132, 7.00039916435363)
(-61.784655184821524, 7.00071781844734)
(8.377589819935137, -10.993377055023)
(-46.07669322153476, -10.9985451171653)
(43.982297482291145, -10.9992875269771)
(72.2566309405206, 7.00030290913574)
(6.283199140682201, -10.9914374751537)
(2.09443126500936, -10.9837511988629)
(87.9645943546813, -10.9997872823013)
(-85.87019930683888, -10.9996615348314)
(-4.188346910456506, -10.9136064362704)
(-33.51032499201856, -10.9966708610988)
(81.68140905955936, -10.9997567832526)
(-70.16223571433554, 7.00053464226723)
(-19.89671530572347, 7.01684369100262)
(-35.604719363084264, -10.9971743870382)
(41.88790242260501, -10.9992276735664)
(-54.454273188059695, -10.9990319883923)
(46.07669254822783, -10.9993406829701)
(-72.25663083746545, 7.00049982489169)
(92.15338455301327, -10.9998045349304)
(-15.707799648568995, 7.04444871776575)
(-2.0942451541035054, -10.9580610926021)
(-99.48376729701715, 7.0002442804138)
(-55.50146972231403, 7.00092490394596)
(-24.085529293069957, 7.00878837290691)
(-90.05898949562697, -10.9996956119311)
(65.9734456082059, 7.00035580746375)
(4.188811730520251, -10.9885020777671)
(-30.368723923667883, 7.00437998594069)
(-79.58701403135224, -10.9995985972209)
(-65.97344545835722, 7.00061614797483)
(-43.982298303789925, -10.998365694161)
(63.87905049541709, 7.00037655167458)
(10.471982200708535, -10.994725153609)
(83.775804157568, -10.9997676404423)
(26.179937645714595, 7.00161599439808)
(21.991146916068807, 7.00208235454299)
(-68.06784058815961, 7.00057322854084)
(-41.88790343610983, -10.9981509117591)
(-98.43656988150887, -10.9997499656211)
(68.06784071989526, 7.00033673139753)
(70.16223583062674, 7.00031914929659)
(85.87019925595533, -10.9997777865355)
(100.53096495245403, -10.9998332921048)
(34.55751859192214, 7.00105415319254)
(37.69911232797407, -10.9990829083288)
(80.63421137356497, 7.00024893295342)
(59.69026026583958, 7.00042387682328)
(96.34217475233129, -10.9998197706834)
(-37.699113932226965, -10.9975717519393)
(-17.802285962008124, 7.02581282137753)
(-28.274326985672133, 7.00538358278082)
(-6.280729345456382, -10.7292819644464)
(56.548667939957234, -10.9995345186815)
(-83.77580421383918, -10.9996423091661)
(-48.171088176676854, -10.9986965369279)
(-29.321537317946852, -10.9951569717402)
(39.79350737056852, -10.9991599484739)
(6.773882217244417e-05, -10.9753088278157)
(-87.96459440080885, -10.9996792510981)
(24.08554231400587, 7.00182706274611)
(-39.79350863668036, -10.9978908295974)
(30.36872817538032, 7.00129040850519)
(-26.179929040993215, 7.00677621251815)
(-50.265483160208234, -10.9988254915811)
Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
$$x_{1} = -92.1533845911891$$
$$x_{2} = 50.2654826946666$$
$$x_{3} = 54.4542728555055$$
$$x_{4} = -94.2477796874062$$
$$x_{5} = -77.4926189422375$$
$$x_{6} = -81.6814091219536$$
$$x_{7} = 98.4365698523018$$
$$x_{8} = 94.2477796525614$$
$$x_{9} = 90.0589894537109$$
$$x_{10} = 48.1710876193121$$
$$x_{11} = -20.9439800058931$$
$$x_{12} = 52.3598777735875$$
$$x_{13} = 8.37758981993514$$
$$x_{14} = -46.0766932215348$$
$$x_{15} = 43.9822974822911$$
$$x_{16} = 6.2831991406822$$
$$x_{17} = 2.09443126500936$$
$$x_{18} = 87.9645943546813$$
$$x_{19} = -85.8701993068389$$
$$x_{20} = -4.18834691045651$$
$$x_{21} = -33.5103249920186$$
$$x_{22} = 81.6814090595594$$
$$x_{23} = -35.6047193630843$$
$$x_{24} = 41.887902422605$$
$$x_{25} = -54.4542731880597$$
$$x_{26} = 46.0766925482278$$
$$x_{27} = 92.1533845530133$$
$$x_{28} = -2.09424515410351$$
$$x_{29} = -90.058989495627$$
$$x_{30} = 4.18881173052025$$
$$x_{31} = -79.5870140313522$$
$$x_{32} = -43.9822983037899$$
$$x_{33} = 10.4719822007085$$
$$x_{34} = 83.775804157568$$
$$x_{35} = -41.8879034361098$$
$$x_{36} = -98.4365698815089$$
$$x_{37} = 85.8701992559553$$
$$x_{38} = 100.530964952454$$
$$x_{39} = 37.6991123279741$$
$$x_{40} = 96.3421747523313$$
$$x_{41} = -37.699113932227$$
$$x_{42} = -6.28072934545638$$
$$x_{43} = 56.5486679399572$$
$$x_{44} = -83.7758042138392$$
$$x_{45} = -48.1710881766769$$
$$x_{46} = -29.3215373179469$$
$$x_{47} = 39.7935073705685$$
$$x_{48} = 6.77388221724442 \cdot 10^{-5}$$
$$x_{49} = -87.9645944008088$$
$$x_{50} = -39.7935086366804$$
$$x_{51} = -50.2654831602082$$
Puntos máximos de la función:
$$x_{51} = 28.274332928754$$
$$x_{51} = 15.7079599940573$$
$$x_{51} = -13.6130651210836$$
$$x_{51} = 17.802355805527$$
$$x_{51} = 78.5398162661312$$
$$x_{51} = 76.4454211581911$$
$$x_{51} = 32.4631233943248$$
$$x_{51} = -21.9911260516957$$
$$x_{51} = -11.516072706381$$
$$x_{51} = -63.8790503242099$$
$$x_{51} = 74.3510260496794$$
$$x_{51} = -57.5958648855072$$
$$x_{51} = 19.8967514261592$$
$$x_{51} = -59.6902600390645$$
$$x_{51} = 61.7846553813613$$
$$x_{51} = -61.7846551848215$$
$$x_{51} = 72.2566309405206$$
$$x_{51} = -70.1622357143355$$
$$x_{51} = -19.8967153057235$$
$$x_{51} = -72.2566308374655$$
$$x_{51} = -15.707799648569$$
$$x_{51} = -99.4837672970172$$
$$x_{51} = -55.501469722314$$
$$x_{51} = -24.08552929307$$
$$x_{51} = 65.9734456082059$$
$$x_{51} = -30.3687239236679$$
$$x_{51} = -65.9734454583572$$
$$x_{51} = 63.8790504954171$$
$$x_{51} = 26.1799376457146$$
$$x_{51} = 21.9911469160688$$
$$x_{51} = -68.0678405881596$$
$$x_{51} = 68.0678407198953$$
$$x_{51} = 70.1622358306267$$
$$x_{51} = 34.5575185919221$$
$$x_{51} = 80.634211373565$$
$$x_{51} = 59.6902602658396$$
$$x_{51} = -17.8022859620081$$
$$x_{51} = -28.2743269856721$$
$$x_{51} = 24.0855423140059$$
$$x_{51} = 30.3687281753803$$
$$x_{51} = -26.1799290409932$$
Decrece en los intervalos
$$\left[100.530964952454, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.4365698815089\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$3 \left(27 \cos{\left(3 x \right)} + \frac{4}{\left(x + 9\right)^{4}}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -53.9306738745076$$
$$x_{2} = -91.629785728643$$
$$x_{3} = -82.2050077706525$$
$$x_{4} = -25.6563406459052$$
$$x_{5} = -89.5353906261352$$
$$x_{6} = -84.2994028728622$$
$$x_{7} = -27.750735506196$$
$$x_{8} = -88.4881930773495$$
$$x_{9} = 31.9395253290756$$
$$x_{10} = -19.373158962253$$
$$x_{11} = 15.1843646367071$$
$$x_{12} = -69.6386371582261$$
$$x_{13} = 12.0427715868985$$
$$x_{14} = -23.5619460001636$$
$$x_{15} = -62.3082542900826$$
$$x_{16} = 18.3259570573732$$
$$x_{17} = 29.8451302307915$$
$$x_{18} = 4.71239037714632$$
$$x_{19} = -97.9129710360917$$
$$x_{20} = 56.0250689862558$$
$$x_{21} = 87.4409955243451$$
$$x_{22} = 26.7035375251233$$
$$x_{23} = 68.5914396020147$$
$$x_{24} = -67.5442420563843$$
$$x_{25} = -100.007366138555$$
$$x_{26} = -38.2227106863921$$
$$x_{27} = 78.0162175650079$$
$$x_{28} = -98.9601685888325$$
$$x_{29} = -47.6474885573097$$
$$x_{30} = -45.5530934493903$$
$$x_{31} = 66.4970444994639$$
$$x_{32} = 16.2315619217049$$
$$x_{33} = -73.8274273621562$$
$$x_{34} = 97.9129710365039$$
$$x_{35} = 20.4203521824188$$
$$x_{36} = 95.8185759340796$$
$$x_{37} = -16.2315439863292$$
$$x_{38} = -75.9218224642154$$
$$x_{39} = 42.4115008305308$$
$$x_{40} = 49.7418836776909$$
$$x_{41} = -36.1283156074593$$
$$x_{42} = 53.9306738834761$$
$$x_{43} = -14.1370960316843$$
$$x_{44} = -71.7330322601555$$
$$x_{45} = -78.0162175663231$$
$$x_{46} = -93.7241808311371$$
$$x_{47} = 14.1371667688346$$
$$x_{48} = 5.75958549099445$$
$$x_{49} = 100.007366138925$$
$$x_{50} = 51.8362787806264$$
$$x_{51} = -7.82779900206498$$
$$x_{52} = -50.7890812492279$$
$$x_{53} = -9.85453033910856$$
$$x_{54} = 75.9218224627028$$
$$x_{55} = 22.5147473006635$$
$$x_{56} = 88.4881930766592$$
$$x_{57} = 86.3937979743157$$
$$x_{58} = 93.724180831652$$
$$x_{59} = -1.57078011608167$$
$$x_{60} = 41.3643032645905$$
$$x_{61} = 71.7330322581294$$
$$x_{62} = -31.9395254898315$$
$$x_{63} = -58.119464082928$$
$$x_{64} = 82.2050077696466$$
$$x_{65} = 34.0339204282884$$
$$x_{66} = -43.4586983396337$$
$$x_{67} = 38.2227106286063$$
$$x_{68} = -49.7418836639154$$
$$x_{69} = 60.2138591916526$$
$$x_{70} = 7.85398102195539$$
$$x_{71} = 1.57079237181025$$
$$x_{72} = -56.0250689789195$$
$$x_{73} = -60.213859186626$$
$$x_{74} = 44.5058959318806$$
$$x_{75} = 27.7507351337812$$
$$x_{76} = 73.8274273604094$$
$$x_{77} = 0.5236047786145$$
$$x_{78} = -5.759138886905$$
$$x_{79} = -34.0339205396254$$
$$x_{80} = -12.0421953022393$$
$$x_{81} = -3.66513046430266$$
$$x_{82} = -21.467551843386$$
$$x_{83} = 9.9483763532899$$
$$x_{84} = 80.1106126673229$$
$$x_{85} = -51.836278769565$$
$$x_{86} = 91.6297857292207$$
$$x_{87} = -95.8185759336195$$
$$x_{88} = -80.110612668471$$
$$x_{89} = -29.8451304706541$$
$$x_{90} = 84.2994028719778$$
$$x_{91} = -65.4498469546506$$
$$x_{92} = 64.4026493968897$$
$$x_{93} = 62.3082542942876$$
$$x_{94} = 58.119464088978$$
$$x_{95} = 40.3171057294171$$
$$x_{96} = 36.128315528189$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -9$$
$$\lim_{x \to -9^-}\left(3 \left(27 \cos{\left(3 x \right)} + \frac{4}{\left(x + 9\right)^{4}}\right)\right) = \infty$$
$$\lim_{x \to -9^+}\left(3 \left(27 \cos{\left(3 x \right)} + \frac{4}{\left(x + 9\right)^{4}}\right)\right) = \infty$$
- los límites son iguales, es decir omitimos el punto correspondiente
Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[100.007366138925, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -98.9601685888325\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$3 \left(27 \cos{\left(3 x \right)} + \frac{4}{\left(x + 9\right)^{4}}\right) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -53.9306738745076$$
$$x_{2} = -91.629785728643$$
$$x_{3} = -82.2050077706525$$
$$x_{4} = -25.6563406459052$$
$$x_{5} = -89.5353906261352$$
$$x_{6} = -84.2994028728622$$
$$x_{7} = -27.750735506196$$
$$x_{8} = -88.4881930773495$$
$$x_{9} = 31.9395253290756$$
$$x_{10} = -19.373158962253$$
$$x_{11} = 15.1843646367071$$
$$x_{12} = -69.6386371582261$$
$$x_{13} = 12.0427715868985$$
$$x_{14} = -23.5619460001636$$
$$x_{15} = -62.3082542900826$$
$$x_{16} = 18.3259570573732$$
$$x_{17} = 29.8451302307915$$
$$x_{18} = 4.71239037714632$$
$$x_{19} = -97.9129710360917$$
$$x_{20} = 56.0250689862558$$
$$x_{21} = 87.4409955243451$$
$$x_{22} = 26.7035375251233$$
$$x_{23} = 68.5914396020147$$
$$x_{24} = -67.5442420563843$$
$$x_{25} = -100.007366138555$$
$$x_{26} = -38.2227106863921$$
$$x_{27} = 78.0162175650079$$
$$x_{28} = -98.9601685888325$$
$$x_{29} = -47.6474885573097$$
$$x_{30} = -45.5530934493903$$
$$x_{31} = 66.4970444994639$$
$$x_{32} = 16.2315619217049$$
$$x_{33} = -73.8274273621562$$
$$x_{34} = 97.9129710365039$$
$$x_{35} = 20.4203521824188$$
$$x_{36} = 95.8185759340796$$
$$x_{37} = -16.2315439863292$$
$$x_{38} = -75.9218224642154$$
$$x_{39} = 42.4115008305308$$
$$x_{40} = 49.7418836776909$$
$$x_{41} = -36.1283156074593$$
$$x_{42} = 53.9306738834761$$
$$x_{43} = -14.1370960316843$$
$$x_{44} = -71.7330322601555$$
$$x_{45} = -78.0162175663231$$
$$x_{46} = -93.7241808311371$$
$$x_{47} = 14.1371667688346$$
$$x_{48} = 5.75958549099445$$
$$x_{49} = 100.007366138925$$
$$x_{50} = 51.8362787806264$$
$$x_{51} = -7.82779900206498$$
$$x_{52} = -50.7890812492279$$
$$x_{53} = -9.85453033910856$$
$$x_{54} = 75.9218224627028$$
$$x_{55} = 22.5147473006635$$
$$x_{56} = 88.4881930766592$$
$$x_{57} = 86.3937979743157$$
$$x_{58} = 93.724180831652$$
$$x_{59} = -1.57078011608167$$
$$x_{60} = 41.3643032645905$$
$$x_{61} = 71.7330322581294$$
$$x_{62} = -31.9395254898315$$
$$x_{63} = -58.119464082928$$
$$x_{64} = 82.2050077696466$$
$$x_{65} = 34.0339204282884$$
$$x_{66} = -43.4586983396337$$
$$x_{67} = 38.2227106286063$$
$$x_{68} = -49.7418836639154$$
$$x_{69} = 60.2138591916526$$
$$x_{70} = 7.85398102195539$$
$$x_{71} = 1.57079237181025$$
$$x_{72} = -56.0250689789195$$
$$x_{73} = -60.213859186626$$
$$x_{74} = 44.5058959318806$$
$$x_{75} = 27.7507351337812$$
$$x_{76} = 73.8274273604094$$
$$x_{77} = 0.5236047786145$$
$$x_{78} = -5.759138886905$$
$$x_{79} = -34.0339205396254$$
$$x_{80} = -12.0421953022393$$
$$x_{81} = -3.66513046430266$$
$$x_{82} = -21.467551843386$$
$$x_{83} = 9.9483763532899$$
$$x_{84} = 80.1106126673229$$
$$x_{85} = -51.836278769565$$
$$x_{86} = 91.6297857292207$$
$$x_{87} = -95.8185759336195$$
$$x_{88} = -80.110612668471$$
$$x_{89} = -29.8451304706541$$
$$x_{90} = 84.2994028719778$$
$$x_{91} = -65.4498469546506$$
$$x_{92} = 64.4026493968897$$
$$x_{93} = 62.3082542942876$$
$$x_{94} = 58.119464088978$$
$$x_{95} = 40.3171057294171$$
$$x_{96} = 36.128315528189$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = -9$$
$$\lim_{x \to -9^-}\left(3 \left(27 \cos{\left(3 x \right)} + \frac{4}{\left(x + 9\right)^{4}}\right)\right) = \infty$$
$$\lim_{x \to -9^+}\left(3 \left(27 \cos{\left(3 x \right)} + \frac{4}{\left(x + 9\right)^{4}}\right)\right) = \infty$$
- los límites son iguales, es decir omitimos el punto correspondiente
Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
$$\left[100.007366138925, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -98.9601685888325\right]$$
Asíntotas verticales
Hay:
$$x_{1} = -9$$
$$x_{1} = -9$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2\right) = \left\langle -11, 7\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -11, 7\right\rangle$$
$$\lim_{x \to \infty}\left(\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2\right) = \left\langle -11, 7\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -11, 7\right\rangle$$
$$\lim_{x \to -\infty}\left(\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2\right) = \left\langle -11, 7\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -11, 7\right\rangle$$
$$\lim_{x \to \infty}\left(\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2\right) = \left\langle -11, 7\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -11, 7\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 2/(x + 9)^2 - 9*cos(3*x) - 2, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
$$\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2 = - 9 \cos{\left(3 x \right)} - 2 + \frac{2}{\left(9 - x\right)^{2}}$$
- No
$$\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2 = 9 \cos{\left(3 x \right)} + 2 - \frac{2}{\left(9 - x\right)^{2}}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2 = - 9 \cos{\left(3 x \right)} - 2 + \frac{2}{\left(9 - x\right)^{2}}$$
- No
$$\left(- 9 \cos{\left(3 x \right)} + \frac{2}{\left(x + 9\right)^{2}}\right) - 2 = 9 \cos{\left(3 x \right)} + 2 - \frac{2}{\left(9 - x\right)^{2}}$$
- No
es decir, función
no es
par ni impar