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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
-x^4+x^2
-
x-e
-
x*(-8)/(x^2+4)
-
(x+5)^2-9
-
Expresiones idénticas
- -cos(x)+(- uno + dos *x)*sin(x)
- menos coseno de (x) más ( menos 1 más 2 multiplicar por x) multiplicar por seno de (x)
- menos coseno de (x) más ( menos uno más dos multiplicar por x) multiplicar por seno de (x)
- -cos(x)+(-1+2x)sin(x)
- -cosx+-1+2xsinx
-
Expresiones semejantes
- -cos(x)+(-1-2*x)*sin(x)
- -cos(x)-(-1+2*x)*sin(x)
- -cos(x)+(1+2*x)*sin(x)
- cos(x)+(-1+2*x)*sin(x)
- -cosx+(-1+2*x)*sinx
-
Expresiones con funciones
- Coseno cos
- cos(2x)/sin(x)
- cos2x+x^3-4
- cos(x+pi/4)+2
- cos(x)*((2*x^3-5)^3)^4
- cos2x-sinx
- Seno sin
- sin(2*x-pi/4)
- sin(5+6*x)^(2)+3^tan(x)
- sin(x)+arcsin(x)
- sin(2*x^3)^2/(x^3)
- sinpiy
Gráfico de la función y = -cos(x)+(-1+2*x)*sin(x)
Solución
Ha introducido
[src]
f(x) = -cos(x) + (-1 + 2*x)*sin(x)
$$f{\left(x \right)} = \left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}$$
f = (2*x - 1)*sin(x) - cos(x)
Gráfico de la función
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:
$$\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 22.01438465735$$
$$x_{2} = 53.4165236742723$$
$$x_{3} = 65.9810813702516$$
$$x_{4} = -31.4315837342855$$
$$x_{5} = 81.6875674938813$$
$$x_{6} = 0.898224447028667$$
$$x_{7} = 25.1530199394671$$
$$x_{8} = 50.2755272166115$$
$$x_{9} = 15.7407582723812$$
$$x_{10} = 34.572192859464$$
$$x_{11} = -81.6874925690229$$
$$x_{12} = 78.5462227120501$$
$$x_{13} = 94.2531127133922$$
$$x_{14} = 12.6076433909519$$
$$x_{15} = -15.7387441015495$$
$$x_{16} = -248.187830183631$$
$$x_{17} = -37.712195924099$$
$$x_{18} = -65.980966533204$$
$$x_{19} = 37.7125473628024$$
$$x_{20} = 43.9937925372228$$
$$x_{21} = 3.31724218377523$$
$$x_{22} = -50.2753294410975$$
$$x_{23} = -25.1522302435149$$
$$x_{24} = 97.3945324653665$$
$$x_{25} = -62.8397468452885$$
$$x_{26} = -72.2635025012724$$
$$x_{27} = 72.2635982404799$$
$$x_{28} = -47.1343860373378$$
$$x_{29} = -59.698566072896$$
$$x_{30} = -0.47418921158996$$
$$x_{31} = -97.3944797571657$$
$$x_{32} = 47.1346110435264$$
$$x_{33} = -78.5461416748096$$
$$x_{34} = -12.6045069275269$$
$$x_{35} = 59.6987063478865$$
$$x_{36} = 87.9703104635367$$
$$x_{37} = 91.1117049489984$$
$$x_{38} = -6.35598540076715$$
$$x_{39} = 62.8398734486626$$
$$x_{40} = -40.8527949886538$$
$$x_{41} = -94.2530564333498$$
$$x_{42} = -18.875356172562$$
$$x_{43} = -100.535913609853$$
$$x_{44} = -22.0133539656788$$
$$x_{45} = 56.5575869273694$$
$$x_{46} = -28.2916982538369$$
$$x_{47} = 40.8530944861043$$
$$x_{48} = -91.1116447213668$$
$$x_{49} = -87.9702458579592$$
$$x_{50} = 31.4320895712942$$
$$x_{51} = -69.1222198708219$$
$$x_{52} = 9.48039736015017$$
$$x_{53} = -75.4048107886474$$
$$x_{54} = -9.47486204798468$$
$$x_{55} = 69.1223245086075$$
$$x_{56} = 100.535963075745$$
$$x_{57} = -53.41634847025$$
$$x_{58} = -56.5574306408327$$
$$x_{59} = -84.8288612619489$$
$$x_{60} = 75.4048987178623$$
$$x_{61} = -34.5717747021855$$
$$x_{62} = 6.36818522389288$$
$$x_{63} = 28.2923225216739$$
$$x_{64} = -3.2733339786093$$
$$x_{65} = 84.8289307407309$$
$$x_{66} = 18.876757492502$$
$$x_{67} = -43.9935342651014$$
$$x_{68} = -122.526177643312$$
o sea hay que resolver la ecuación:
$$\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 22.01438465735$$
$$x_{2} = 53.4165236742723$$
$$x_{3} = 65.9810813702516$$
$$x_{4} = -31.4315837342855$$
$$x_{5} = 81.6875674938813$$
$$x_{6} = 0.898224447028667$$
$$x_{7} = 25.1530199394671$$
$$x_{8} = 50.2755272166115$$
$$x_{9} = 15.7407582723812$$
$$x_{10} = 34.572192859464$$
$$x_{11} = -81.6874925690229$$
$$x_{12} = 78.5462227120501$$
$$x_{13} = 94.2531127133922$$
$$x_{14} = 12.6076433909519$$
$$x_{15} = -15.7387441015495$$
$$x_{16} = -248.187830183631$$
$$x_{17} = -37.712195924099$$
$$x_{18} = -65.980966533204$$
$$x_{19} = 37.7125473628024$$
$$x_{20} = 43.9937925372228$$
$$x_{21} = 3.31724218377523$$
$$x_{22} = -50.2753294410975$$
$$x_{23} = -25.1522302435149$$
$$x_{24} = 97.3945324653665$$
$$x_{25} = -62.8397468452885$$
$$x_{26} = -72.2635025012724$$
$$x_{27} = 72.2635982404799$$
$$x_{28} = -47.1343860373378$$
$$x_{29} = -59.698566072896$$
$$x_{30} = -0.47418921158996$$
$$x_{31} = -97.3944797571657$$
$$x_{32} = 47.1346110435264$$
$$x_{33} = -78.5461416748096$$
$$x_{34} = -12.6045069275269$$
$$x_{35} = 59.6987063478865$$
$$x_{36} = 87.9703104635367$$
$$x_{37} = 91.1117049489984$$
$$x_{38} = -6.35598540076715$$
$$x_{39} = 62.8398734486626$$
$$x_{40} = -40.8527949886538$$
$$x_{41} = -94.2530564333498$$
$$x_{42} = -18.875356172562$$
$$x_{43} = -100.535913609853$$
$$x_{44} = -22.0133539656788$$
$$x_{45} = 56.5575869273694$$
$$x_{46} = -28.2916982538369$$
$$x_{47} = 40.8530944861043$$
$$x_{48} = -91.1116447213668$$
$$x_{49} = -87.9702458579592$$
$$x_{50} = 31.4320895712942$$
$$x_{51} = -69.1222198708219$$
$$x_{52} = 9.48039736015017$$
$$x_{53} = -75.4048107886474$$
$$x_{54} = -9.47486204798468$$
$$x_{55} = 69.1223245086075$$
$$x_{56} = 100.535963075745$$
$$x_{57} = -53.41634847025$$
$$x_{58} = -56.5574306408327$$
$$x_{59} = -84.8288612619489$$
$$x_{60} = 75.4048987178623$$
$$x_{61} = -34.5717747021855$$
$$x_{62} = 6.36818522389288$$
$$x_{63} = 28.2923225216739$$
$$x_{64} = -3.2733339786093$$
$$x_{65} = 84.8289307407309$$
$$x_{66} = 18.876757492502$$
$$x_{67} = -43.9935342651014$$
$$x_{68} = -122.526177643312$$
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
$$\left(2 x - 1\right) \cos{\left(x \right)} + 3 \sin{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -45.5856300973313$$
$$x_{2} = -83.27010955499$$
$$x_{3} = -2.09492138588594$$
$$x_{4} = -76.9883753378549$$
$$x_{5} = 45.5863507026718$$
$$x_{6} = 55.0053847101118$$
$$x_{7} = 105.257671677194$$
$$x_{8} = 58.1454793419602$$
$$x_{9} = -67.5662758307315$$
$$x_{10} = -11.1239094230235$$
$$x_{11} = 76.9886282578302$$
$$x_{12} = -58.1450361281976$$
$$x_{13} = -4.97958645877522$$
$$x_{14} = 5.03201855014322$$
$$x_{15} = 8.0501007584565$$
$$x_{16} = 80.1294476912726$$
$$x_{17} = -20.4916878532928$$
$$x_{18} = 39.3085402260932$$
$$x_{19} = -17.3625376671531$$
$$x_{20} = 42.4472447951378$$
$$x_{21} = -89.5520461210964$$
$$x_{22} = 2.27298232005305$$
$$x_{23} = 23.6267142359735$$
$$x_{24} = 36.1703424960528$$
$$x_{25} = 61.2857285828291$$
$$x_{26} = 17.3674549261486$$
$$x_{27} = 29.8961131447549$$
$$x_{28} = 33.032797548069$$
$$x_{29} = 92.6932520167225$$
$$x_{30} = -55.0048895078955$$
$$x_{31} = -8.02809006801769$$
$$x_{32} = -92.6930775072709$$
$$x_{33} = 48.7257798022585$$
$$x_{34} = -61.2853295847371$$
$$x_{35} = -95.834145479277$$
$$x_{36} = -23.6240435868063$$
$$x_{37} = -51.8649160864964$$
$$x_{38} = -39.3075716244322$$
$$x_{39} = 14.2458606217585$$
$$x_{40} = 11.1356848370861$$
$$x_{41} = 51.8654729829429$$
$$x_{42} = -70.7068969658595$$
$$x_{43} = -48.7251489329418$$
$$x_{44} = 86.4112560772636$$
$$x_{45} = 0.198412792932205$$
$$x_{46} = 73.8478749993387$$
$$x_{47} = 98.9753996408205$$
$$x_{48} = 20.495229888127$$
$$x_{49} = 95.8343087410424$$
$$x_{50} = -98.9752465735786$$
$$x_{51} = 89.5522330815776$$
$$x_{52} = -26.7585107959123$$
$$x_{53} = 70.7071967867599$$
$$x_{54} = -86.4110552844808$$
$$x_{55} = 64.4261096842988$$
$$x_{56} = -64.4257486040079$$
$$x_{57} = -73.8476001224597$$
$$x_{58} = 83.2703257734076$$
$$x_{59} = 26.7605953575094$$
$$x_{60} = -80.1292141995248$$
$$x_{61} = -36.1691989884899$$
$$x_{62} = -42.4464138779966$$
$$x_{63} = -33.0314272117789$$
$$x_{64} = -29.8944413312634$$
$$x_{65} = -14.2385913566217$$
$$x_{66} = 67.5666041499633$$
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:
$$x_{1} = 55.0053847101118$$
$$x_{2} = 105.257671677194$$
$$x_{3} = -67.5662758307315$$
$$x_{4} = -11.1239094230235$$
$$x_{5} = -4.97958645877522$$
$$x_{6} = 5.03201855014322$$
$$x_{7} = 80.1294476912726$$
$$x_{8} = -17.3625376671531$$
$$x_{9} = 42.4472447951378$$
$$x_{10} = 23.6267142359735$$
$$x_{11} = 36.1703424960528$$
$$x_{12} = 61.2857285828291$$
$$x_{13} = 17.3674549261486$$
$$x_{14} = 29.8961131447549$$
$$x_{15} = 92.6932520167225$$
$$x_{16} = -55.0048895078955$$
$$x_{17} = -92.6930775072709$$
$$x_{18} = 48.7257798022585$$
$$x_{19} = -61.2853295847371$$
$$x_{20} = -23.6240435868063$$
$$x_{21} = 11.1356848370861$$
$$x_{22} = -48.7251489329418$$
$$x_{23} = 86.4112560772636$$
$$x_{24} = 0.198412792932205$$
$$x_{25} = 73.8478749993387$$
$$x_{26} = 98.9753996408205$$
$$x_{27} = -98.9752465735786$$
$$x_{28} = -86.4110552844808$$
$$x_{29} = -73.8476001224597$$
$$x_{30} = -80.1292141995248$$
$$x_{31} = -36.1691989884899$$
$$x_{32} = -42.4464138779966$$
$$x_{33} = -29.8944413312634$$
$$x_{34} = 67.5666041499633$$
Puntos máximos de la función:
$$x_{34} = -45.5856300973313$$
$$x_{34} = -83.27010955499$$
$$x_{34} = -2.09492138588594$$
$$x_{34} = -76.9883753378549$$
$$x_{34} = 45.5863507026718$$
$$x_{34} = 58.1454793419602$$
$$x_{34} = 76.9886282578302$$
$$x_{34} = -58.1450361281976$$
$$x_{34} = 8.0501007584565$$
$$x_{34} = -20.4916878532928$$
$$x_{34} = 39.3085402260932$$
$$x_{34} = -89.5520461210964$$
$$x_{34} = 2.27298232005305$$
$$x_{34} = 33.032797548069$$
$$x_{34} = -8.02809006801769$$
$$x_{34} = -95.834145479277$$
$$x_{34} = -51.8649160864964$$
$$x_{34} = -39.3075716244322$$
$$x_{34} = 14.2458606217585$$
$$x_{34} = 51.8654729829429$$
$$x_{34} = -70.7068969658595$$
$$x_{34} = 20.495229888127$$
$$x_{34} = 95.8343087410424$$
$$x_{34} = 89.5522330815776$$
$$x_{34} = -26.7585107959123$$
$$x_{34} = 70.7071967867599$$
$$x_{34} = 64.4261096842988$$
$$x_{34} = -64.4257486040079$$
$$x_{34} = 83.2703257734076$$
$$x_{34} = 26.7605953575094$$
$$x_{34} = -33.0314272117789$$
$$x_{34} = -14.2385913566217$$
Decrece en los intervalos
$$\left[105.257671677194, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9752465735786\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\left(2 x - 1\right) \cos{\left(x \right)} + 3 \sin{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -45.5856300973313$$
$$x_{2} = -83.27010955499$$
$$x_{3} = -2.09492138588594$$
$$x_{4} = -76.9883753378549$$
$$x_{5} = 45.5863507026718$$
$$x_{6} = 55.0053847101118$$
$$x_{7} = 105.257671677194$$
$$x_{8} = 58.1454793419602$$
$$x_{9} = -67.5662758307315$$
$$x_{10} = -11.1239094230235$$
$$x_{11} = 76.9886282578302$$
$$x_{12} = -58.1450361281976$$
$$x_{13} = -4.97958645877522$$
$$x_{14} = 5.03201855014322$$
$$x_{15} = 8.0501007584565$$
$$x_{16} = 80.1294476912726$$
$$x_{17} = -20.4916878532928$$
$$x_{18} = 39.3085402260932$$
$$x_{19} = -17.3625376671531$$
$$x_{20} = 42.4472447951378$$
$$x_{21} = -89.5520461210964$$
$$x_{22} = 2.27298232005305$$
$$x_{23} = 23.6267142359735$$
$$x_{24} = 36.1703424960528$$
$$x_{25} = 61.2857285828291$$
$$x_{26} = 17.3674549261486$$
$$x_{27} = 29.8961131447549$$
$$x_{28} = 33.032797548069$$
$$x_{29} = 92.6932520167225$$
$$x_{30} = -55.0048895078955$$
$$x_{31} = -8.02809006801769$$
$$x_{32} = -92.6930775072709$$
$$x_{33} = 48.7257798022585$$
$$x_{34} = -61.2853295847371$$
$$x_{35} = -95.834145479277$$
$$x_{36} = -23.6240435868063$$
$$x_{37} = -51.8649160864964$$
$$x_{38} = -39.3075716244322$$
$$x_{39} = 14.2458606217585$$
$$x_{40} = 11.1356848370861$$
$$x_{41} = 51.8654729829429$$
$$x_{42} = -70.7068969658595$$
$$x_{43} = -48.7251489329418$$
$$x_{44} = 86.4112560772636$$
$$x_{45} = 0.198412792932205$$
$$x_{46} = 73.8478749993387$$
$$x_{47} = 98.9753996408205$$
$$x_{48} = 20.495229888127$$
$$x_{49} = 95.8343087410424$$
$$x_{50} = -98.9752465735786$$
$$x_{51} = 89.5522330815776$$
$$x_{52} = -26.7585107959123$$
$$x_{53} = 70.7071967867599$$
$$x_{54} = -86.4110552844808$$
$$x_{55} = 64.4261096842988$$
$$x_{56} = -64.4257486040079$$
$$x_{57} = -73.8476001224597$$
$$x_{58} = 83.2703257734076$$
$$x_{59} = 26.7605953575094$$
$$x_{60} = -80.1292141995248$$
$$x_{61} = -36.1691989884899$$
$$x_{62} = -42.4464138779966$$
$$x_{63} = -33.0314272117789$$
$$x_{64} = -29.8944413312634$$
$$x_{65} = -14.2385913566217$$
$$x_{66} = 67.5666041499633$$
Signos de extremos en los puntos:
(-45.58563009733132, 92.1550076713403)
(-83.27010955499001, 167.531269623105)
(-2.0949213858859403, 4.99362510094841)
(-76.98837533785488, 154.967076336648)
(45.58635070267175, 90.1560896516624)
(55.00538471011178, -108.997022329722)
(105.25767167719371, -209.508185808685)
(58.145479341960204, 115.277959128447)
(-67.56627583073154, -136.121539675013)
(-11.123909423023479, -23.1846198986047)
(76.98862825783024, 152.967455848603)
(-58.14503612819762, 117.277293902343)
(-4.97958645877522, -10.8343120289872)
(5.032018550143224, -8.91917555623417)
(8.050100758456498, 15.0055980979256)
(80.12944769127256, -159.249480932697)
(-20.491687853292817, 41.9478742761831)
(39.30854022609317, 77.5977908493213)
(-17.362537667153074, -35.6834557894953)
(42.4472447951378, -83.8766385351894)
(-89.55204612109635, 180.095766612943)
(2.2729823200530452, 3.35299133763984)
(23.62671423597348, -46.2211683346969)
(36.17034249605284, -71.3197055241561)
(61.28572858282906, -121.559128129836)
(17.367454926148618, -33.6908819676733)
(29.89611314475495, -58.7667955576171)
(33.03279754806898, 65.0426025825689)
(92.69325201672254, -184.378371639055)
(-55.00488950789551, -110.996279020194)
(-8.028090068017692, 16.9715442883252)
(-92.69307750727087, -186.378109812048)
(48.72577980225852, -96.4360265469457)
(-61.2853295847371, -123.558529304126)
(-95.83414547927696, 192.660507916009)
(-23.624043586806327, -48.2171475834257)
(-51.86491608649638, 104.715524285881)
(-39.307571624432185, 79.5963360086251)
(14.245860621758535, 27.437962905159)
(11.135684837086066, -21.2025748245251)
(51.865472982942904, 102.716360270807)
(-70.70689696585954, 142.403267084147)
(-48.72514893294176, -98.4350794212227)
(86.41125607726357, -171.813785541528)
(0.19841279293220518, -1.09927450005257)
(73.84787499933874, -146.685530097094)
(98.97539964082053, -196.94318537473)
(20.495229888126993, 39.9532133611104)
(95.83430874104242, 190.660752863647)
(-98.97524657357857, -198.942955725531)
(89.55223308157761, 178.096047125781)
(-26.758510795912326, 54.489611113495)
(70.70719678675987, 140.403717000996)
(-86.41105528448084, -173.81348426917)
(64.42610968429884, 127.840495143801)
(-64.42574860400788, 129.839953254293)
(-73.84760012245974, -148.685117625865)
(83.27032577340765, 165.53159404719)
(26.7605953575094, 52.4927469501143)
(-80.12921419952481, -161.249130582586)
(-36.16919898848991, -73.3179875606176)
(-42.446413877996626, -85.8753907334664)
(-33.031427211778926, 67.0405431960901)
(-29.89444133126343, -60.7642820563578)
(-14.23859135662174, 29.4269486603258)
(67.56660414996325, -134.122032376309)
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:
$$x_{1} = 55.0053847101118$$
$$x_{2} = 105.257671677194$$
$$x_{3} = -67.5662758307315$$
$$x_{4} = -11.1239094230235$$
$$x_{5} = -4.97958645877522$$
$$x_{6} = 5.03201855014322$$
$$x_{7} = 80.1294476912726$$
$$x_{8} = -17.3625376671531$$
$$x_{9} = 42.4472447951378$$
$$x_{10} = 23.6267142359735$$
$$x_{11} = 36.1703424960528$$
$$x_{12} = 61.2857285828291$$
$$x_{13} = 17.3674549261486$$
$$x_{14} = 29.8961131447549$$
$$x_{15} = 92.6932520167225$$
$$x_{16} = -55.0048895078955$$
$$x_{17} = -92.6930775072709$$
$$x_{18} = 48.7257798022585$$
$$x_{19} = -61.2853295847371$$
$$x_{20} = -23.6240435868063$$
$$x_{21} = 11.1356848370861$$
$$x_{22} = -48.7251489329418$$
$$x_{23} = 86.4112560772636$$
$$x_{24} = 0.198412792932205$$
$$x_{25} = 73.8478749993387$$
$$x_{26} = 98.9753996408205$$
$$x_{27} = -98.9752465735786$$
$$x_{28} = -86.4110552844808$$
$$x_{29} = -73.8476001224597$$
$$x_{30} = -80.1292141995248$$
$$x_{31} = -36.1691989884899$$
$$x_{32} = -42.4464138779966$$
$$x_{33} = -29.8944413312634$$
$$x_{34} = 67.5666041499633$$
Puntos máximos de la función:
$$x_{34} = -45.5856300973313$$
$$x_{34} = -83.27010955499$$
$$x_{34} = -2.09492138588594$$
$$x_{34} = -76.9883753378549$$
$$x_{34} = 45.5863507026718$$
$$x_{34} = 58.1454793419602$$
$$x_{34} = 76.9886282578302$$
$$x_{34} = -58.1450361281976$$
$$x_{34} = 8.0501007584565$$
$$x_{34} = -20.4916878532928$$
$$x_{34} = 39.3085402260932$$
$$x_{34} = -89.5520461210964$$
$$x_{34} = 2.27298232005305$$
$$x_{34} = 33.032797548069$$
$$x_{34} = -8.02809006801769$$
$$x_{34} = -95.834145479277$$
$$x_{34} = -51.8649160864964$$
$$x_{34} = -39.3075716244322$$
$$x_{34} = 14.2458606217585$$
$$x_{34} = 51.8654729829429$$
$$x_{34} = -70.7068969658595$$
$$x_{34} = 20.495229888127$$
$$x_{34} = 95.8343087410424$$
$$x_{34} = 89.5522330815776$$
$$x_{34} = -26.7585107959123$$
$$x_{34} = 70.7071967867599$$
$$x_{34} = 64.4261096842988$$
$$x_{34} = -64.4257486040079$$
$$x_{34} = 83.2703257734076$$
$$x_{34} = 26.7605953575094$$
$$x_{34} = -33.0314272117789$$
$$x_{34} = -14.2385913566217$$
Decrece en los intervalos
$$\left[105.257671677194, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9752465735786\right]$$
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
$$- \left(2 x - 1\right) \sin{\left(x \right)} + 5 \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -40.9010162530929$$
$$x_{2} = -22.1013137713123$$
$$x_{3} = -91.1334627976286$$
$$x_{4} = -1.02349513473659$$
$$x_{5} = -9.66591283377341$$
$$x_{6} = 100.555945738604$$
$$x_{7} = 34.6306365858486$$
$$x_{8} = 84.8526304609639$$
$$x_{9} = -87.9928376622495$$
$$x_{10} = -69.1509162590798$$
$$x_{11} = -81.7118088818525$$
$$x_{12} = -34.6285665560111$$
$$x_{13} = -62.8712826711762$$
$$x_{14} = 75.4315750870618$$
$$x_{15} = -66.0110157970615$$
$$x_{16} = -6.62082455533636$$
$$x_{17} = 22.1063430830515$$
$$x_{18} = -25.2296015255588$$
$$x_{19} = -12.7528187647602$$
$$x_{20} = 12.7674096589572$$
$$x_{21} = 31.4964068352961$$
$$x_{22} = 91.1337635011842$$
$$x_{23} = -84.8522836452833$$
$$x_{24} = 94.2744330129804$$
$$x_{25} = 9.69037504695292$$
$$x_{26} = 40.9025030598333$$
$$x_{27} = -75.4311363666522$$
$$x_{28} = 25.2334766791702$$
$$x_{29} = 50.3156254469131$$
$$x_{30} = 78.5718271958884$$
$$x_{31} = 6.66825325197955$$
$$x_{32} = 28.3638163783154$$
$$x_{33} = -28.3607410560874$$
$$x_{34} = 37.7660965820389$$
$$x_{35} = 81.7121828338092$$
$$x_{36} = 47.1773977914817$$
$$x_{37} = 44.0396530959431$$
$$x_{38} = 62.8719137709601$$
$$x_{39} = -97.4148990910485$$
$$x_{40} = -18.9772130423993$$
$$x_{41} = 66.0115884045148$$
$$x_{42} = 59.7324419829608$$
$$x_{43} = -47.1762788004668$$
$$x_{44} = -15.8596055393397$$
$$x_{45} = -78.5714227960697$$
$$x_{46} = 18.9839922940528$$
$$x_{47} = -50.3146412391001$$
$$x_{48} = -59.7317429606586$$
$$x_{49} = -56.5924284528966$$
$$x_{50} = 69.1514381376921$$
$$x_{51} = 97.4151622991313$$
$$x_{52} = -37.7643540751581$$
$$x_{53} = -317.308724217129$$
$$x_{54} = -94.2741519923053$$
$$x_{55} = -100.55569870239$$
$$x_{56} = 53.4542506462294$$
$$x_{57} = 72.291440059835$$
$$x_{58} = -31.493907955455$$
$$x_{59} = -3.68054615295438$$
$$x_{60} = 1.27147686651553$$
$$x_{61} = 3.79120904399595$$
$$x_{62} = 87.9931601901129$$
$$x_{63} = 56.5932069664086$$
$$x_{64} = 15.8692138210114$$
$$x_{65} = -72.2909624609804$$
$$x_{66} = -53.4533782929539$$
$$x_{67} = -44.0383696863753$$
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
$$\left[97.4151622991313, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.55569870239\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \left(2 x - 1\right) \sin{\left(x \right)} + 5 \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -40.9010162530929$$
$$x_{2} = -22.1013137713123$$
$$x_{3} = -91.1334627976286$$
$$x_{4} = -1.02349513473659$$
$$x_{5} = -9.66591283377341$$
$$x_{6} = 100.555945738604$$
$$x_{7} = 34.6306365858486$$
$$x_{8} = 84.8526304609639$$
$$x_{9} = -87.9928376622495$$
$$x_{10} = -69.1509162590798$$
$$x_{11} = -81.7118088818525$$
$$x_{12} = -34.6285665560111$$
$$x_{13} = -62.8712826711762$$
$$x_{14} = 75.4315750870618$$
$$x_{15} = -66.0110157970615$$
$$x_{16} = -6.62082455533636$$
$$x_{17} = 22.1063430830515$$
$$x_{18} = -25.2296015255588$$
$$x_{19} = -12.7528187647602$$
$$x_{20} = 12.7674096589572$$
$$x_{21} = 31.4964068352961$$
$$x_{22} = 91.1337635011842$$
$$x_{23} = -84.8522836452833$$
$$x_{24} = 94.2744330129804$$
$$x_{25} = 9.69037504695292$$
$$x_{26} = 40.9025030598333$$
$$x_{27} = -75.4311363666522$$
$$x_{28} = 25.2334766791702$$
$$x_{29} = 50.3156254469131$$
$$x_{30} = 78.5718271958884$$
$$x_{31} = 6.66825325197955$$
$$x_{32} = 28.3638163783154$$
$$x_{33} = -28.3607410560874$$
$$x_{34} = 37.7660965820389$$
$$x_{35} = 81.7121828338092$$
$$x_{36} = 47.1773977914817$$
$$x_{37} = 44.0396530959431$$
$$x_{38} = 62.8719137709601$$
$$x_{39} = -97.4148990910485$$
$$x_{40} = -18.9772130423993$$
$$x_{41} = 66.0115884045148$$
$$x_{42} = 59.7324419829608$$
$$x_{43} = -47.1762788004668$$
$$x_{44} = -15.8596055393397$$
$$x_{45} = -78.5714227960697$$
$$x_{46} = 18.9839922940528$$
$$x_{47} = -50.3146412391001$$
$$x_{48} = -59.7317429606586$$
$$x_{49} = -56.5924284528966$$
$$x_{50} = 69.1514381376921$$
$$x_{51} = 97.4151622991313$$
$$x_{52} = -37.7643540751581$$
$$x_{53} = -317.308724217129$$
$$x_{54} = -94.2741519923053$$
$$x_{55} = -100.55569870239$$
$$x_{56} = 53.4542506462294$$
$$x_{57} = 72.291440059835$$
$$x_{58} = -31.493907955455$$
$$x_{59} = -3.68054615295438$$
$$x_{60} = 1.27147686651553$$
$$x_{61} = 3.79120904399595$$
$$x_{62} = 87.9931601901129$$
$$x_{63} = 56.5932069664086$$
$$x_{64} = 15.8692138210114$$
$$x_{65} = -72.2909624609804$$
$$x_{66} = -53.4533782929539$$
$$x_{67} = -44.0383696863753$$
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
$$\left[97.4151622991313, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.55569870239\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, \infty\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función -cos(x) + (-1 + 2*x)*sin(x), dividida por x con x->+oo y x ->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{x}\right)$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{x}\right)$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{x}\right)$$
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:
$$\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)} = - \left(- 2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}$$
- No
$$\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)} = \left(- 2 x - 1\right) \sin{\left(x \right)} + \cos{\left(x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)} = - \left(- 2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}$$
- No
$$\left(2 x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)} = \left(- 2 x - 1\right) \sin{\left(x \right)} + \cos{\left(x \right)}$$
- No
es decir, función
no es
par ni impar