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- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
y=(x+1)^3
-
2*x^2-6*x
-
y=5x
-
y=4^x
-
Expresiones idénticas
- x*sin(x)+cos(x)-x
- x multiplicar por seno de (x) más coseno de (x) menos x
- xsin(x)+cos(x)-x
- xsinx+cosx-x
-
Expresiones semejantes
- x*sin(x)+cos(x)+x
- x*sin(x)-cos(x)-x
- x*sinx+cosx-x
-
Expresiones con funciones
- Seno sin
- sin(0,5x+pi/4)+0,5
- sin(x×4)
- sin(x^4-3*x^2+1)
- sin(x)+cos(x)+sin(x)/x!
- sin(x*(pi/180))
- Coseno cos
- cos[x]/x^2
- cos(x-pi/3)-1
- cos(x)/(1+cos^2*2x)
- cos(1)/((3*x))
- cos(sqrt(x))+1
Gráfico de la función y = x*sin(x)+cos(x)-x
Solución
Ha introducido
[src]
f(x) = x*sin(x) + cos(x) - x
$$f{\left(x \right)} = - x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)$$
f = -x + x*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:
$$- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 7.85398163397448$$
$$x_{2} = -73.8274273593601$$
$$x_{3} = -54.9778714378214$$
$$x_{4} = 64.3715822869017$$
$$x_{5} = 39.2189234266452$$
$$x_{6} = -92.6553987604331$$
$$x_{7} = 89.5130484454873$$
$$x_{8} = -10.8111042087213$$
$$x_{9} = -4.71238898038469$$
$$x_{10} = 14.1371669411541$$
$$x_{11} = 45.5091533451563$$
$$x_{12} = 20.3220161353369$$
$$x_{13} = 58.1194640914112$$
$$x_{14} = 58.0850352160434$$
$$x_{15} = -23.4768059032848$$
$$x_{16} = 70.6858347057703$$
$$x_{17} = -36.1283155162826$$
$$x_{18} = 51.7976718062027$$
$$x_{19} = 26.6284652377851$$
$$x_{20} = 95.7976993646524$$
$$x_{21} = -36.0728864084812$$
$$x_{22} = -29.7779917432681$$
$$x_{23} = 7.59205618191083$$
$$x_{24} = -92.6769832808989$$
$$x_{25} = -86.3937979737193$$
$$x_{26} = -98.9399549958912$$
$$x_{27} = 83.2281761528687$$
$$x_{28} = -10.9955742875643$$
$$x_{29} = 39.2699081698724$$
$$x_{30} = -67.5146210051587$$
$$x_{31} = 102.08216980829$$
$$x_{32} = -61.261056745001$$
$$x_{33} = -67.5442420521806$$
$$x_{34} = -86.3706429922226$$
$$x_{35} = 26.7035375555132$$
$$x_{36} = -48.6946861306418$$
$$x_{37} = 51.8362787842316$$
$$x_{38} = -80.0856406984281$$
$$x_{39} = 76.9430282181184$$
$$x_{40} = -98.9601685880785$$
$$x_{41} = -42.4115008234622$$
$$x_{42} = 95.8185759344887$$
$$x_{43} = 1.5707963267949$$
$$x_{44} = 45.553093477052$$
$$x_{45} = -17.2787595947439$$
$$x_{46} = -42.3643000278463$$
$$x_{47} = -54.9414730837878$$
$$x_{48} = 76.9690200129499$$
$$x_{49} = 32.9259992567895$$
$$x_{50} = 70.6575310493539$$
$$x_{51} = 20.4203522483337$$
$$x_{52} = -80.1106126665397$$
$$x_{53} = -4.2502319840436$$
$$x_{54} = 32.9867228626928$$
$$x_{55} = 64.4026493985908$$
$$x_{56} = -23.5619449019235$$
$$x_{57} = -73.8003288675086$$
$$x_{58} = -48.6535849776189$$
$$x_{59} = 89.5353906273091$$
$$x_{60} = -61.2283950657729$$
$$x_{61} = 83.2522053201295$$
$$x_{62} = -29.845130209103$$
$$x_{63} = -17.1623570970183$$
$$x_{64} = 13.9944961126907$$
$$x_{65} = 133.502707088111$$
o sea hay que resolver la ecuación:
$$- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 7.85398163397448$$
$$x_{2} = -73.8274273593601$$
$$x_{3} = -54.9778714378214$$
$$x_{4} = 64.3715822869017$$
$$x_{5} = 39.2189234266452$$
$$x_{6} = -92.6553987604331$$
$$x_{7} = 89.5130484454873$$
$$x_{8} = -10.8111042087213$$
$$x_{9} = -4.71238898038469$$
$$x_{10} = 14.1371669411541$$
$$x_{11} = 45.5091533451563$$
$$x_{12} = 20.3220161353369$$
$$x_{13} = 58.1194640914112$$
$$x_{14} = 58.0850352160434$$
$$x_{15} = -23.4768059032848$$
$$x_{16} = 70.6858347057703$$
$$x_{17} = -36.1283155162826$$
$$x_{18} = 51.7976718062027$$
$$x_{19} = 26.6284652377851$$
$$x_{20} = 95.7976993646524$$
$$x_{21} = -36.0728864084812$$
$$x_{22} = -29.7779917432681$$
$$x_{23} = 7.59205618191083$$
$$x_{24} = -92.6769832808989$$
$$x_{25} = -86.3937979737193$$
$$x_{26} = -98.9399549958912$$
$$x_{27} = 83.2281761528687$$
$$x_{28} = -10.9955742875643$$
$$x_{29} = 39.2699081698724$$
$$x_{30} = -67.5146210051587$$
$$x_{31} = 102.08216980829$$
$$x_{32} = -61.261056745001$$
$$x_{33} = -67.5442420521806$$
$$x_{34} = -86.3706429922226$$
$$x_{35} = 26.7035375555132$$
$$x_{36} = -48.6946861306418$$
$$x_{37} = 51.8362787842316$$
$$x_{38} = -80.0856406984281$$
$$x_{39} = 76.9430282181184$$
$$x_{40} = -98.9601685880785$$
$$x_{41} = -42.4115008234622$$
$$x_{42} = 95.8185759344887$$
$$x_{43} = 1.5707963267949$$
$$x_{44} = 45.553093477052$$
$$x_{45} = -17.2787595947439$$
$$x_{46} = -42.3643000278463$$
$$x_{47} = -54.9414730837878$$
$$x_{48} = 76.9690200129499$$
$$x_{49} = 32.9259992567895$$
$$x_{50} = 70.6575310493539$$
$$x_{51} = 20.4203522483337$$
$$x_{52} = -80.1106126665397$$
$$x_{53} = -4.2502319840436$$
$$x_{54} = 32.9867228626928$$
$$x_{55} = 64.4026493985908$$
$$x_{56} = -23.5619449019235$$
$$x_{57} = -73.8003288675086$$
$$x_{58} = -48.6535849776189$$
$$x_{59} = 89.5353906273091$$
$$x_{60} = -61.2283950657729$$
$$x_{61} = 83.2522053201295$$
$$x_{62} = -29.845130209103$$
$$x_{63} = -17.1623570970183$$
$$x_{64} = 13.9944961126907$$
$$x_{65} = 133.502707088111$$
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
$$x \cos{\left(x \right)} - 1 = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -2.07393280909122$$
$$x_{2} = 32.9563750616135$$
$$x_{3} = 20.3712437074438$$
$$x_{4} = -64.4181735917203$$
$$x_{5} = -33.0170149091969$$
$$x_{6} = 4.91718592528713$$
$$x_{7} = -70.699979453112$$
$$x_{8} = -83.2642155700859$$
$$x_{9} = 7.72415319239641$$
$$x_{10} = -10.9037335277384$$
$$x_{11} = -54.959675275262$$
$$x_{12} = -26.740942117298$$
$$x_{13} = 11.0859017287718$$
$$x_{14} = 98.9702728040995$$
$$x_{15} = 14.0660135689384$$
$$x_{16} = 95.8081382182729$$
$$x_{17} = -67.5294331532335$$
$$x_{18} = -14.2076100006438$$
$$x_{19} = -80.0981276558536$$
$$x_{20} = -29.8115799030901$$
$$x_{21} = 89.5242202334874$$
$$x_{22} = -58.1366657885594$$
$$x_{23} = -23.5194140147849$$
$$x_{24} = -7.97963107097301$$
$$x_{25} = -86.3822212589452$$
$$x_{26} = -36.1006116108761$$
$$x_{27} = -95.829011377113$$
$$x_{28} = -20.4692255293053$$
$$x_{29} = -42.3879070002498$$
$$x_{30} = 61.2773767058956$$
$$x_{31} = -45.5750370742992$$
$$x_{32} = 67.5590444598741$$
$$x_{33} = 17.336473487102$$
$$x_{34} = -114.676852122197$$
$$x_{35} = 23.6043227065406$$
$$x_{36} = 29.8786052250774$$
$$x_{37} = -17.2206571155732$$
$$x_{38} = 83.240191603726$$
$$x_{39} = 86.405371586641$$
$$x_{40} = 58.1022522048044$$
$$x_{41} = 92.6877723998433$$
$$x_{42} = 39.2444240846477$$
$$x_{43} = 54.9960555621608$$
$$x_{44} = -51.8555643132686$$
$$x_{45} = 45.5311287148944$$
$$x_{46} = -76.9820104261667$$
$$x_{47} = -92.6661916492115$$
$$x_{48} = -61.2447280834131$$
$$x_{49} = 36.1559769880743$$
$$x_{50} = -4.48766960334109$$
$$x_{51} = -48.6741398947227$$
$$x_{52} = 70.671684294851$$
$$x_{53} = 80.1230937867295$$
$$x_{54} = 51.8169788924771$$
$$x_{55} = -39.2953592151719$$
$$x_{56} = 73.8409703906111$$
$$x_{57} = -73.8138793572668$$
$$x_{58} = 26.6660278619112$$
$$x_{59} = -98.9500623082067$$
$$x_{60} = -89.5465582344838$$
$$x_{61} = 64.3871177170664$$
$$x_{62} = 42.4350684201498$$
$$x_{63} = 48.7152150401823$$
$$x_{64} = 76.9560252131026$$
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:
$$x_{1} = 4.91718592528713$$
$$x_{2} = -10.9037335277384$$
$$x_{3} = -54.959675275262$$
$$x_{4} = 11.0859017287718$$
$$x_{5} = 98.9702728040995$$
$$x_{6} = -67.5294331532335$$
$$x_{7} = -80.0981276558536$$
$$x_{8} = -29.8115799030901$$
$$x_{9} = -23.5194140147849$$
$$x_{10} = -86.3822212589452$$
$$x_{11} = -36.1006116108761$$
$$x_{12} = -42.3879070002498$$
$$x_{13} = 61.2773767058956$$
$$x_{14} = 67.5590444598741$$
$$x_{15} = 17.336473487102$$
$$x_{16} = 23.6043227065406$$
$$x_{17} = 29.8786052250774$$
$$x_{18} = -17.2206571155732$$
$$x_{19} = 86.405371586641$$
$$x_{20} = 92.6877723998433$$
$$x_{21} = 54.9960555621608$$
$$x_{22} = -92.6661916492115$$
$$x_{23} = -61.2447280834131$$
$$x_{24} = 36.1559769880743$$
$$x_{25} = -4.48766960334109$$
$$x_{26} = -48.6741398947227$$
$$x_{27} = 80.1230937867295$$
$$x_{28} = 73.8409703906111$$
$$x_{29} = -73.8138793572668$$
$$x_{30} = -98.9500623082067$$
$$x_{31} = 42.4350684201498$$
$$x_{32} = 48.7152150401823$$
Puntos máximos de la función:
$$x_{32} = -2.07393280909122$$
$$x_{32} = 32.9563750616135$$
$$x_{32} = 20.3712437074438$$
$$x_{32} = -64.4181735917203$$
$$x_{32} = -33.0170149091969$$
$$x_{32} = -70.699979453112$$
$$x_{32} = -83.2642155700859$$
$$x_{32} = 7.72415319239641$$
$$x_{32} = -26.740942117298$$
$$x_{32} = 14.0660135689384$$
$$x_{32} = 95.8081382182729$$
$$x_{32} = -14.2076100006438$$
$$x_{32} = 89.5242202334874$$
$$x_{32} = -58.1366657885594$$
$$x_{32} = -7.97963107097301$$
$$x_{32} = -95.829011377113$$
$$x_{32} = -20.4692255293053$$
$$x_{32} = -45.5750370742992$$
$$x_{32} = -114.676852122197$$
$$x_{32} = 83.240191603726$$
$$x_{32} = 58.1022522048044$$
$$x_{32} = 39.2444240846477$$
$$x_{32} = -51.8555643132686$$
$$x_{32} = 45.5311287148944$$
$$x_{32} = -76.9820104261667$$
$$x_{32} = 70.671684294851$$
$$x_{32} = 51.8169788924771$$
$$x_{32} = -39.2953592151719$$
$$x_{32} = 26.6660278619112$$
$$x_{32} = -89.5465582344838$$
$$x_{32} = 64.3871177170664$$
$$x_{32} = 76.9560252131026$$
Decrece en los intervalos
$$\left[98.9702728040995, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9500623082067\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$x \cos{\left(x \right)} - 1 = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -2.07393280909122$$
$$x_{2} = 32.9563750616135$$
$$x_{3} = 20.3712437074438$$
$$x_{4} = -64.4181735917203$$
$$x_{5} = -33.0170149091969$$
$$x_{6} = 4.91718592528713$$
$$x_{7} = -70.699979453112$$
$$x_{8} = -83.2642155700859$$
$$x_{9} = 7.72415319239641$$
$$x_{10} = -10.9037335277384$$
$$x_{11} = -54.959675275262$$
$$x_{12} = -26.740942117298$$
$$x_{13} = 11.0859017287718$$
$$x_{14} = 98.9702728040995$$
$$x_{15} = 14.0660135689384$$
$$x_{16} = 95.8081382182729$$
$$x_{17} = -67.5294331532335$$
$$x_{18} = -14.2076100006438$$
$$x_{19} = -80.0981276558536$$
$$x_{20} = -29.8115799030901$$
$$x_{21} = 89.5242202334874$$
$$x_{22} = -58.1366657885594$$
$$x_{23} = -23.5194140147849$$
$$x_{24} = -7.97963107097301$$
$$x_{25} = -86.3822212589452$$
$$x_{26} = -36.1006116108761$$
$$x_{27} = -95.829011377113$$
$$x_{28} = -20.4692255293053$$
$$x_{29} = -42.3879070002498$$
$$x_{30} = 61.2773767058956$$
$$x_{31} = -45.5750370742992$$
$$x_{32} = 67.5590444598741$$
$$x_{33} = 17.336473487102$$
$$x_{34} = -114.676852122197$$
$$x_{35} = 23.6043227065406$$
$$x_{36} = 29.8786052250774$$
$$x_{37} = -17.2206571155732$$
$$x_{38} = 83.240191603726$$
$$x_{39} = 86.405371586641$$
$$x_{40} = 58.1022522048044$$
$$x_{41} = 92.6877723998433$$
$$x_{42} = 39.2444240846477$$
$$x_{43} = 54.9960555621608$$
$$x_{44} = -51.8555643132686$$
$$x_{45} = 45.5311287148944$$
$$x_{46} = -76.9820104261667$$
$$x_{47} = -92.6661916492115$$
$$x_{48} = -61.2447280834131$$
$$x_{49} = 36.1559769880743$$
$$x_{50} = -4.48766960334109$$
$$x_{51} = -48.6741398947227$$
$$x_{52} = 70.671684294851$$
$$x_{53} = 80.1230937867295$$
$$x_{54} = 51.8169788924771$$
$$x_{55} = -39.2953592151719$$
$$x_{56} = 73.8409703906111$$
$$x_{57} = -73.8138793572668$$
$$x_{58} = 26.6660278619112$$
$$x_{59} = -98.9500623082067$$
$$x_{60} = -89.5465582344838$$
$$x_{61} = 64.3871177170664$$
$$x_{62} = 42.4350684201498$$
$$x_{63} = 48.7152150401823$$
$$x_{64} = 76.9560252131026$$
Signos de extremos en los puntos:
(-2.073932809091215, 3.40867683863142)
(32.95637506161347, 0.0151680777319427)
(20.371243707443842, 0.0245295981859854)
(-64.41817359172032, 128.813061361402)
(-33.017014909196874, 65.9885952228472)
(4.917185925287132, -9.52824562032736)
(-70.69997945311201, 141.378742138954)
(-83.26421557008594, 166.510415981788)
(7.724153192396411, 0.0644584754526054)
(-10.903733527738439, -0.0457590213602597)
(-54.959675275261986, -0.00909682613061591)
(-26.740942117297966, 53.4257839325997)
(11.085901728771786, -22.0364043437832)
(98.97027280409945, -197.925389413115)
(14.066013568938363, 0.0355016446467076)
(95.80813821827292, 0.00521862120285732)
(-67.52943315323353, -0.00740377288414606)
(-14.20761000064383, 28.3095990846546)
(-80.09812765585362, -0.00624209990532165)
(-29.811579903090074, -0.0167672854758187)
(89.52422023348744, 0.00558490653321542)
(-58.13666578855936, 116.247529667622)
(-23.519414014784864, -0.0212494164164099)
(-7.979631070973006, 15.7710355605792)
(-86.3822212589452, -0.00578803415562845)
(-36.100611610876136, -0.0138475229953983)
(-95.82901137711305, 191.642369732338)
(-20.46922552930527, 40.8651557259926)
(-42.3879070002498, -0.0117941753995794)
(61.27737670589561, -122.530274013769)
(-45.57503707429922, 91.1171600734531)
(67.55904445987407, -135.095885713621)
(17.336473487101994, -34.5864001575149)
(-114.67685212219666, 229.340623928028)
(23.60432270654059, -47.1450882176196)
(29.87860522507741, -59.7070026145832)
(-17.220657115573236, -0.0290103781662836)
(83.240191603726, 0.00600649696690425)
(86.40537158664104, -172.79338294768)
(58.10225220480441, 0.00860488101699275)
(92.68777239984328, -185.359361278257)
(39.2444240846477, 0.0127385946678444)
(54.99605556216085, -109.964835689388)
(-51.855564313268616, 103.682201229561)
(45.53112871489442, 0.0109801734091377)
(-76.9820104261667, 153.944535506522)
(-92.66619164921153, -0.00539555401387304)
(-61.24472808341312, -0.00816342379199142)
(36.15597698807427, -72.2704644139298)
(-4.487669603341088, -0.109997879424804)
(-48.67413989472275, -0.0102713109855017)
(70.671684294851, 0.0070746151713621)
(80.12309378672954, -160.227466136207)
(51.816978892477124, 0.0096484481933814)
(-39.29535921517187, 78.5525439230185)
(73.84097039061113, -147.661626544837)
(-73.81387935726681, -0.00677348298323466)
(26.666027861911218, 0.0187438522602434)
(-98.9500623082067, -0.00505292488966802)
(-89.54655823448383, 179.0763652323)
(64.38711771706639, 0.00776506019175827)
(42.43506842014976, -84.8347870814508)
(48.71521504018234, -97.3996377974613)
(76.95602521310259, 0.00649694276592072)
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:
$$x_{1} = 4.91718592528713$$
$$x_{2} = -10.9037335277384$$
$$x_{3} = -54.959675275262$$
$$x_{4} = 11.0859017287718$$
$$x_{5} = 98.9702728040995$$
$$x_{6} = -67.5294331532335$$
$$x_{7} = -80.0981276558536$$
$$x_{8} = -29.8115799030901$$
$$x_{9} = -23.5194140147849$$
$$x_{10} = -86.3822212589452$$
$$x_{11} = -36.1006116108761$$
$$x_{12} = -42.3879070002498$$
$$x_{13} = 61.2773767058956$$
$$x_{14} = 67.5590444598741$$
$$x_{15} = 17.336473487102$$
$$x_{16} = 23.6043227065406$$
$$x_{17} = 29.8786052250774$$
$$x_{18} = -17.2206571155732$$
$$x_{19} = 86.405371586641$$
$$x_{20} = 92.6877723998433$$
$$x_{21} = 54.9960555621608$$
$$x_{22} = -92.6661916492115$$
$$x_{23} = -61.2447280834131$$
$$x_{24} = 36.1559769880743$$
$$x_{25} = -4.48766960334109$$
$$x_{26} = -48.6741398947227$$
$$x_{27} = 80.1230937867295$$
$$x_{28} = 73.8409703906111$$
$$x_{29} = -73.8138793572668$$
$$x_{30} = -98.9500623082067$$
$$x_{31} = 42.4350684201498$$
$$x_{32} = 48.7152150401823$$
Puntos máximos de la función:
$$x_{32} = -2.07393280909122$$
$$x_{32} = 32.9563750616135$$
$$x_{32} = 20.3712437074438$$
$$x_{32} = -64.4181735917203$$
$$x_{32} = -33.0170149091969$$
$$x_{32} = -70.699979453112$$
$$x_{32} = -83.2642155700859$$
$$x_{32} = 7.72415319239641$$
$$x_{32} = -26.740942117298$$
$$x_{32} = 14.0660135689384$$
$$x_{32} = 95.8081382182729$$
$$x_{32} = -14.2076100006438$$
$$x_{32} = 89.5242202334874$$
$$x_{32} = -58.1366657885594$$
$$x_{32} = -7.97963107097301$$
$$x_{32} = -95.829011377113$$
$$x_{32} = -20.4692255293053$$
$$x_{32} = -45.5750370742992$$
$$x_{32} = -114.676852122197$$
$$x_{32} = 83.240191603726$$
$$x_{32} = 58.1022522048044$$
$$x_{32} = 39.2444240846477$$
$$x_{32} = -51.8555643132686$$
$$x_{32} = 45.5311287148944$$
$$x_{32} = -76.9820104261667$$
$$x_{32} = 70.671684294851$$
$$x_{32} = 51.8169788924771$$
$$x_{32} = -39.2953592151719$$
$$x_{32} = 26.6660278619112$$
$$x_{32} = -89.5465582344838$$
$$x_{32} = 64.3871177170664$$
$$x_{32} = 76.9560252131026$$
Decrece en los intervalos
$$\left[98.9702728040995, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9500623082067\right]$$
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
$$- x \sin{\left(x \right)} + \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 87.9759605524932$$
$$x_{2} = 34.5864242152889$$
$$x_{3} = 62.8477631944545$$
$$x_{4} = -69.1295029738953$$
$$x_{5} = 50.2853663377737$$
$$x_{6} = 18.90240995686$$
$$x_{7} = 56.5663442798215$$
$$x_{8} = 6.43729817917195$$
$$x_{9} = -62.8477631944545$$
$$x_{10} = 37.7256128277765$$
$$x_{11} = -9.52933440536196$$
$$x_{12} = 15.7712848748159$$
$$x_{13} = 59.7070073053355$$
$$x_{14} = 12.6452872238566$$
$$x_{15} = 100.540910786842$$
$$x_{16} = 78.5525459842429$$
$$x_{17} = 31.4477146375462$$
$$x_{18} = -72.270467060309$$
$$x_{19} = -15.7712848748159$$
$$x_{20} = 44.0050179208308$$
$$x_{21} = -94.2583883450399$$
$$x_{22} = -0.86033358901938$$
$$x_{23} = -34.5864242152889$$
$$x_{24} = -31.4477146375462$$
$$x_{25} = 84.8347887180423$$
$$x_{26} = -75.4114834888481$$
$$x_{27} = -28.309642854452$$
$$x_{28} = -59.7070073053355$$
$$x_{29} = -81.6936492356017$$
$$x_{30} = 28.309642854452$$
$$x_{31} = -22.0364967279386$$
$$x_{32} = -25.1724463266467$$
$$x_{33} = 81.6936492356017$$
$$x_{34} = 65.9885986984904$$
$$x_{35} = -65.9885986984904$$
$$x_{36} = -84.8347887180423$$
$$x_{37} = -78.5525459842429$$
$$x_{38} = 3.42561845948173$$
$$x_{39} = -50.2853663377737$$
$$x_{40} = 9.52933440536196$$
$$x_{41} = 91.1171613944647$$
$$x_{42} = -100.540910786842$$
$$x_{43} = -47.145097736761$$
$$x_{44} = -147.661626855354$$
$$x_{45} = 75.4114834888481$$
$$x_{46} = 94.2583883450399$$
$$x_{47} = -87.9759605524932$$
$$x_{48} = 47.145097736761$$
$$x_{49} = -116.247530303932$$
$$x_{50} = -91.1171613944647$$
$$x_{51} = -12.6452872238566$$
$$x_{52} = 25.1724463266467$$
$$x_{53} = 72.270467060309$$
$$x_{54} = 40.8651703304881$$
$$x_{55} = -40.8651703304881$$
$$x_{56} = -44.0050179208308$$
$$x_{57} = 0.86033358901938$$
$$x_{58} = 69.1295029738953$$
$$x_{59} = -3.42561845948173$$
$$x_{60} = -97.3996388790738$$
$$x_{61} = -56.5663442798215$$
$$x_{62} = 97.3996388790738$$
$$x_{63} = -6.43729817917195$$
$$x_{64} = -53.4257904773947$$
$$x_{65} = 53.4257904773947$$
$$x_{66} = 22.0364967279386$$
$$x_{67} = -37.7256128277765$$
$$x_{68} = -18.90240995686$$
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
$$\left[97.3996388790738, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.540910786842\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- x \sin{\left(x \right)} + \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 87.9759605524932$$
$$x_{2} = 34.5864242152889$$
$$x_{3} = 62.8477631944545$$
$$x_{4} = -69.1295029738953$$
$$x_{5} = 50.2853663377737$$
$$x_{6} = 18.90240995686$$
$$x_{7} = 56.5663442798215$$
$$x_{8} = 6.43729817917195$$
$$x_{9} = -62.8477631944545$$
$$x_{10} = 37.7256128277765$$
$$x_{11} = -9.52933440536196$$
$$x_{12} = 15.7712848748159$$
$$x_{13} = 59.7070073053355$$
$$x_{14} = 12.6452872238566$$
$$x_{15} = 100.540910786842$$
$$x_{16} = 78.5525459842429$$
$$x_{17} = 31.4477146375462$$
$$x_{18} = -72.270467060309$$
$$x_{19} = -15.7712848748159$$
$$x_{20} = 44.0050179208308$$
$$x_{21} = -94.2583883450399$$
$$x_{22} = -0.86033358901938$$
$$x_{23} = -34.5864242152889$$
$$x_{24} = -31.4477146375462$$
$$x_{25} = 84.8347887180423$$
$$x_{26} = -75.4114834888481$$
$$x_{27} = -28.309642854452$$
$$x_{28} = -59.7070073053355$$
$$x_{29} = -81.6936492356017$$
$$x_{30} = 28.309642854452$$
$$x_{31} = -22.0364967279386$$
$$x_{32} = -25.1724463266467$$
$$x_{33} = 81.6936492356017$$
$$x_{34} = 65.9885986984904$$
$$x_{35} = -65.9885986984904$$
$$x_{36} = -84.8347887180423$$
$$x_{37} = -78.5525459842429$$
$$x_{38} = 3.42561845948173$$
$$x_{39} = -50.2853663377737$$
$$x_{40} = 9.52933440536196$$
$$x_{41} = 91.1171613944647$$
$$x_{42} = -100.540910786842$$
$$x_{43} = -47.145097736761$$
$$x_{44} = -147.661626855354$$
$$x_{45} = 75.4114834888481$$
$$x_{46} = 94.2583883450399$$
$$x_{47} = -87.9759605524932$$
$$x_{48} = 47.145097736761$$
$$x_{49} = -116.247530303932$$
$$x_{50} = -91.1171613944647$$
$$x_{51} = -12.6452872238566$$
$$x_{52} = 25.1724463266467$$
$$x_{53} = 72.270467060309$$
$$x_{54} = 40.8651703304881$$
$$x_{55} = -40.8651703304881$$
$$x_{56} = -44.0050179208308$$
$$x_{57} = 0.86033358901938$$
$$x_{58} = 69.1295029738953$$
$$x_{59} = -3.42561845948173$$
$$x_{60} = -97.3996388790738$$
$$x_{61} = -56.5663442798215$$
$$x_{62} = 97.3996388790738$$
$$x_{63} = -6.43729817917195$$
$$x_{64} = -53.4257904773947$$
$$x_{65} = 53.4257904773947$$
$$x_{66} = 22.0364967279386$$
$$x_{67} = -37.7256128277765$$
$$x_{68} = -18.90240995686$$
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
$$\left[97.3996388790738, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.540910786842\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)\right) = \left\langle 0, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle 0, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)\right) = \left\langle -\infty, 0\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, 0\right\rangle$$
$$\lim_{x \to -\infty}\left(- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)\right) = \left\langle 0, \infty\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle 0, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)\right) = \left\langle -\infty, 0\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -\infty, 0\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x*sin(x) + cos(x) - x, dividida por x con x->+oo y x ->-oo
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)}{x}\right) = \left\langle -\infty, 0\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -\infty, 0\right\rangle x$$
True
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)}{x}\right)$$
$$\lim_{x \to \infty}\left(\frac{- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)}{x}\right) = \left\langle -\infty, 0\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = \left\langle -\infty, 0\right\rangle x$$
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:
$$- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) = x \sin{\left(x \right)} + x + \cos{\left(x \right)}$$
- No
$$- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) = - x \sin{\left(x \right)} - x - \cos{\left(x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) = x \sin{\left(x \right)} + x + \cos{\left(x \right)}$$
- No
$$- x + \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) = - x \sin{\left(x \right)} - x - \cos{\left(x \right)}$$
- No
es decir, función
no es
par ni impar