Sr Examen
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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
-sqrt(-1+x^2)
-
(e^x)*sin(x)
-
-exp(-2*x)-exp(3*x)
-
-exp(-3*x)-exp(2*x)
-
Expresiones idénticas
- sin(x)+|sin(x)|
- seno de (x) más módulo de seno de (x)|
- sinx+|sinx|
-
Expresiones semejantes
- sin(x)-|sin(x)|
- sin(x)+|sinx|
- sinx+|sinx|
-
Expresiones con funciones
- Seno sin
- sin((x+2)^(1/2)-2^(1/2))
- sin(sqrt(x+2)-sqrt2)
- sin^2(0.6x)cos(0.3x^2)-0.2
- sin(x)^2+cos(x)^2-10*x
- sin(3*x)^(5)
- Seno sin
- sin((x+2)^(1/2)-2^(1/2))
- sin(sqrt(x+2)-sqrt2)
- sin^2(0.6x)cos(0.3x^2)-0.2
- sin(x)^2+cos(x)^2-10*x
- sin(3*x)^(5)
- Módulo |
- |4*x*x+x-2|+|x+4|-1
- |sin(x)/x|
- |1-sqrt(x-2)|
- |-sqrt(x)-1|
- |2x-4|+x
Gráfico de la función y = sin(x)+|sin(x)|
Solución
Ha introducido
[src]
f(x) = sin(x) + |sin(x)|
$$f{\left(x \right)} = \sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right|$$
f = sin(x) + Abs(sin(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:
$$\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right| = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 43.982069633651$$
$$x_{2} = -8$$
$$x_{3} = 60$$
$$x_{4} = 0$$
$$x_{5} = -6.30522058058903$$
$$x_{6} = 16$$
$$x_{7} = -97.3499667521246$$
$$x_{8} = -63.2864355002418$$
$$x_{9} = 62$$
$$x_{10} = -56.6690796528042$$
$$x_{11} = -88$$
$$x_{12} = -44$$
$$x_{13} = -90.91267565216$$
$$x_{14} = 48$$
$$x_{15} = 42$$
$$x_{16} = -65.75$$
$$x_{17} = 22$$
$$x_{18} = -58$$
$$x_{19} = -70$$
$$x_{20} = 24$$
$$x_{21} = 24.6017779691209$$
$$x_{22} = 92$$
$$x_{23} = -100.640646664$$
$$x_{24} = 34.5842665569944$$
$$x_{25} = -21.75$$
$$x_{26} = 18$$
$$x_{27} = 94$$
$$x_{28} = 10$$
$$x_{29} = -26$$
$$x_{30} = 31.2715540931915$$
$$x_{31} = -84.1327162871329$$
$$x_{32} = 237.820176516025$$
$$x_{33} = -75.7312933201993$$
$$x_{34} = 40.9899632144942$$
$$x_{35} = 66$$
$$x_{36} = -70.7721096888483$$
$$x_{37} = -78$$
$$x_{38} = -94.7701122835159$$
$$x_{39} = -76$$
$$x_{40} = 54$$
$$x_{41} = 47.6705111225119$$
$$x_{42} = -32$$
$$x_{43} = 98$$
$$x_{44} = 86$$
$$x_{45} = -53.3722767275529$$
$$x_{46} = -34$$
$$x_{47} = 28.2746695216654$$
$$x_{48} = 55.4742156532357$$
$$x_{49} = -40.2021993477909$$
$$x_{50} = 3.73233523118589$$
$$x_{51} = 99.2691857797929$$
$$x_{52} = -63.1691856934172$$
$$x_{53} = -15.7072713012488$$
$$x_{54} = 91.6119970620733$$
$$x_{55} = 6$$
$$x_{56} = -50.2844729179922$$
$$x_{57} = -96$$
$$x_{58} = 100$$
$$x_{59} = 16.8553768214997$$
$$x_{60} = -46.9461956376453$$
$$x_{61} = -40$$
$$x_{62} = -26.9481791435702$$
$$x_{63} = 84.9595043176841$$
$$x_{64} = -38$$
$$x_{65} = -32.0322958555426$$
$$x_{66} = 74$$
$$x_{67} = 72.256706298959$$
$$x_{68} = -82$$
$$x_{69} = -64$$
$$x_{70} = 81.6523025788846$$
$$x_{71} = -12.6983278279058$$
$$x_{72} = 68.5412303252988$$
$$x_{73} = -46$$
$$x_{74} = 75.2405069639506$$
$$x_{75} = -28$$
$$x_{76} = -52$$
$$x_{77} = -2$$
$$x_{78} = 68$$
$$x_{79} = 36$$
$$x_{80} = 56$$
$$x_{81} = 12$$
$$x_{82} = 4$$
$$x_{83} = -59.6891039826377$$
$$x_{84} = -90$$
$$x_{85} = 60.6346576021357$$
$$x_{86} = 80$$
$$x_{87} = 50$$
$$x_{88} = -19.3408705762922$$
$$x_{89} = -72$$
$$x_{90} = -20$$
$$x_{91} = 11.7052375414856$$
$$x_{92} = -94.2640254685101$$
$$x_{93} = -14$$
$$x_{94} = 78.5630734444415$$
$$x_{95} = -9.39419101904358$$
$$x_{96} = 87.9640478120565$$
$$x_{97} = -2.97862544745825$$
$$x_{98} = 37.6737115819271$$
$$x_{99} = 1085430.64119322$$
$$x_{100} = 30$$
$$x_{101} = -84$$
o sea hay que resolver la ecuación:
$$\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right| = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 43.982069633651$$
$$x_{2} = -8$$
$$x_{3} = 60$$
$$x_{4} = 0$$
$$x_{5} = -6.30522058058903$$
$$x_{6} = 16$$
$$x_{7} = -97.3499667521246$$
$$x_{8} = -63.2864355002418$$
$$x_{9} = 62$$
$$x_{10} = -56.6690796528042$$
$$x_{11} = -88$$
$$x_{12} = -44$$
$$x_{13} = -90.91267565216$$
$$x_{14} = 48$$
$$x_{15} = 42$$
$$x_{16} = -65.75$$
$$x_{17} = 22$$
$$x_{18} = -58$$
$$x_{19} = -70$$
$$x_{20} = 24$$
$$x_{21} = 24.6017779691209$$
$$x_{22} = 92$$
$$x_{23} = -100.640646664$$
$$x_{24} = 34.5842665569944$$
$$x_{25} = -21.75$$
$$x_{26} = 18$$
$$x_{27} = 94$$
$$x_{28} = 10$$
$$x_{29} = -26$$
$$x_{30} = 31.2715540931915$$
$$x_{31} = -84.1327162871329$$
$$x_{32} = 237.820176516025$$
$$x_{33} = -75.7312933201993$$
$$x_{34} = 40.9899632144942$$
$$x_{35} = 66$$
$$x_{36} = -70.7721096888483$$
$$x_{37} = -78$$
$$x_{38} = -94.7701122835159$$
$$x_{39} = -76$$
$$x_{40} = 54$$
$$x_{41} = 47.6705111225119$$
$$x_{42} = -32$$
$$x_{43} = 98$$
$$x_{44} = 86$$
$$x_{45} = -53.3722767275529$$
$$x_{46} = -34$$
$$x_{47} = 28.2746695216654$$
$$x_{48} = 55.4742156532357$$
$$x_{49} = -40.2021993477909$$
$$x_{50} = 3.73233523118589$$
$$x_{51} = 99.2691857797929$$
$$x_{52} = -63.1691856934172$$
$$x_{53} = -15.7072713012488$$
$$x_{54} = 91.6119970620733$$
$$x_{55} = 6$$
$$x_{56} = -50.2844729179922$$
$$x_{57} = -96$$
$$x_{58} = 100$$
$$x_{59} = 16.8553768214997$$
$$x_{60} = -46.9461956376453$$
$$x_{61} = -40$$
$$x_{62} = -26.9481791435702$$
$$x_{63} = 84.9595043176841$$
$$x_{64} = -38$$
$$x_{65} = -32.0322958555426$$
$$x_{66} = 74$$
$$x_{67} = 72.256706298959$$
$$x_{68} = -82$$
$$x_{69} = -64$$
$$x_{70} = 81.6523025788846$$
$$x_{71} = -12.6983278279058$$
$$x_{72} = 68.5412303252988$$
$$x_{73} = -46$$
$$x_{74} = 75.2405069639506$$
$$x_{75} = -28$$
$$x_{76} = -52$$
$$x_{77} = -2$$
$$x_{78} = 68$$
$$x_{79} = 36$$
$$x_{80} = 56$$
$$x_{81} = 12$$
$$x_{82} = 4$$
$$x_{83} = -59.6891039826377$$
$$x_{84} = -90$$
$$x_{85} = 60.6346576021357$$
$$x_{86} = 80$$
$$x_{87} = 50$$
$$x_{88} = -19.3408705762922$$
$$x_{89} = -72$$
$$x_{90} = -20$$
$$x_{91} = 11.7052375414856$$
$$x_{92} = -94.2640254685101$$
$$x_{93} = -14$$
$$x_{94} = 78.5630734444415$$
$$x_{95} = -9.39419101904358$$
$$x_{96} = 87.9640478120565$$
$$x_{97} = -2.97862544745825$$
$$x_{98} = 37.6737115819271$$
$$x_{99} = 1085430.64119322$$
$$x_{100} = 30$$
$$x_{101} = -84$$
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
$$\cos{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} + \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 74$$
$$x_{2} = -98.9601685880785$$
$$x_{3} = -4.71238898038469$$
$$x_{4} = -88$$
$$x_{5} = -86.3937979737193$$
$$x_{6} = 12$$
$$x_{7} = 94$$
$$x_{8} = 39.2699081698724$$
$$x_{9} = -34$$
$$x_{10} = -21.7750019507615$$
$$x_{11} = -96$$
$$x_{12} = 60$$
$$x_{13} = -26$$
$$x_{14} = 54$$
$$x_{15} = 10$$
$$x_{16} = -46$$
$$x_{17} = -54.9778714378214$$
$$x_{18} = 26.7035375555132$$
$$x_{19} = -82$$
$$x_{20} = 20.4203522483337$$
$$x_{21} = -92.6769832808989$$
$$x_{22} = -64$$
$$x_{23} = 18$$
$$x_{24} = -28$$
$$x_{25} = 76.9690200129499$$
$$x_{26} = 58.1194640914112$$
$$x_{27} = -29.845130209103$$
$$x_{28} = -278.382090328623$$
$$x_{29} = 86$$
$$x_{30} = 66$$
$$x_{31} = 4$$
$$x_{32} = -44$$
$$x_{33} = -90$$
$$x_{34} = 42$$
$$x_{35} = -31.9859091024297$$
$$x_{36} = 50$$
$$x_{37} = 7.85398163397448$$
$$x_{38} = -8$$
$$x_{39} = 89.5353906273091$$
$$x_{40} = -58$$
$$x_{41} = 36$$
$$x_{42} = 51.8362787842316$$
$$x_{43} = 70.6858347057703$$
$$x_{44} = -84$$
$$x_{45} = 45.553093477052$$
$$x_{46} = 56$$
$$x_{47} = 16$$
$$x_{48} = -38$$
$$x_{49} = -70$$
$$x_{50} = 80$$
$$x_{51} = 14.1371669411541$$
$$x_{52} = -67.5442420521806$$
$$x_{53} = 24$$
$$x_{54} = 100$$
$$x_{55} = -32$$
$$x_{56} = -76$$
$$x_{57} = 92$$
$$x_{58} = 48$$
$$x_{59} = -0.266574298765612$$
$$x_{60} = 83.2522053201295$$
$$x_{61} = -72$$
$$x_{62} = -65.75$$
$$x_{63} = -40$$
$$x_{64} = 68$$
$$x_{65} = -20$$
$$x_{66} = -65.1984042550898$$
$$x_{67} = -2$$
$$x_{68} = -17.2787595947439$$
$$x_{69} = 95.8185759344887$$
$$x_{70} = -52$$
$$x_{71} = -78$$
$$x_{72} = -36.1283155162826$$
$$x_{73} = 1.5707963267949$$
$$x_{74} = 62.6154246585311$$
$$x_{75} = 30$$
$$x_{76} = 62$$
$$x_{77} = -61.261056745001$$
$$x_{78} = -73.8274273593601$$
$$x_{79} = 6$$
$$x_{80} = 64.4026493985908$$
$$x_{81} = -14$$
$$x_{82} = -2350.21477648167$$
$$x_{83} = -48.6946861306418$$
$$x_{84} = -21.75$$
$$x_{85} = 98$$
$$x_{86} = -23.5619449019235$$
$$x_{87} = -42.4115008234622$$
$$x_{88} = 32.9867228626928$$
$$x_{89} = -10.9955742875643$$
$$x_{90} = -80.1106126665397$$
$$x_{91} = 22$$
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:
La función no tiene puntos mínimos
Puntos máximos de la función:
$$x_{91} = -98.9601685880785$$
$$x_{91} = -4.71238898038469$$
$$x_{91} = -86.3937979737193$$
$$x_{91} = 39.2699081698724$$
$$x_{91} = -54.9778714378214$$
$$x_{91} = 26.7035375555132$$
$$x_{91} = 20.4203522483337$$
$$x_{91} = -92.6769832808989$$
$$x_{91} = 76.9690200129499$$
$$x_{91} = 58.1194640914112$$
$$x_{91} = -29.845130209103$$
$$x_{91} = 7.85398163397448$$
$$x_{91} = 89.5353906273091$$
$$x_{91} = 51.8362787842316$$
$$x_{91} = 70.6858347057703$$
$$x_{91} = 45.553093477052$$
$$x_{91} = 14.1371669411541$$
$$x_{91} = -67.5442420521806$$
$$x_{91} = 83.2522053201295$$
$$x_{91} = -17.2787595947439$$
$$x_{91} = 95.8185759344887$$
$$x_{91} = -36.1283155162826$$
$$x_{91} = 1.5707963267949$$
$$x_{91} = -61.261056745001$$
$$x_{91} = -73.8274273593601$$
$$x_{91} = 64.4026493985908$$
$$x_{91} = -48.6946861306418$$
$$x_{91} = -23.5619449019235$$
$$x_{91} = -42.4115008234622$$
$$x_{91} = 32.9867228626928$$
$$x_{91} = -10.9955742875643$$
$$x_{91} = -80.1106126665397$$
Decrece en los intervalos
$$\left(-\infty, -98.9601685880785\right]$$
Crece en los intervalos
$$\left[95.8185759344887, \infty\right)$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\cos{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} + \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 74$$
$$x_{2} = -98.9601685880785$$
$$x_{3} = -4.71238898038469$$
$$x_{4} = -88$$
$$x_{5} = -86.3937979737193$$
$$x_{6} = 12$$
$$x_{7} = 94$$
$$x_{8} = 39.2699081698724$$
$$x_{9} = -34$$
$$x_{10} = -21.7750019507615$$
$$x_{11} = -96$$
$$x_{12} = 60$$
$$x_{13} = -26$$
$$x_{14} = 54$$
$$x_{15} = 10$$
$$x_{16} = -46$$
$$x_{17} = -54.9778714378214$$
$$x_{18} = 26.7035375555132$$
$$x_{19} = -82$$
$$x_{20} = 20.4203522483337$$
$$x_{21} = -92.6769832808989$$
$$x_{22} = -64$$
$$x_{23} = 18$$
$$x_{24} = -28$$
$$x_{25} = 76.9690200129499$$
$$x_{26} = 58.1194640914112$$
$$x_{27} = -29.845130209103$$
$$x_{28} = -278.382090328623$$
$$x_{29} = 86$$
$$x_{30} = 66$$
$$x_{31} = 4$$
$$x_{32} = -44$$
$$x_{33} = -90$$
$$x_{34} = 42$$
$$x_{35} = -31.9859091024297$$
$$x_{36} = 50$$
$$x_{37} = 7.85398163397448$$
$$x_{38} = -8$$
$$x_{39} = 89.5353906273091$$
$$x_{40} = -58$$
$$x_{41} = 36$$
$$x_{42} = 51.8362787842316$$
$$x_{43} = 70.6858347057703$$
$$x_{44} = -84$$
$$x_{45} = 45.553093477052$$
$$x_{46} = 56$$
$$x_{47} = 16$$
$$x_{48} = -38$$
$$x_{49} = -70$$
$$x_{50} = 80$$
$$x_{51} = 14.1371669411541$$
$$x_{52} = -67.5442420521806$$
$$x_{53} = 24$$
$$x_{54} = 100$$
$$x_{55} = -32$$
$$x_{56} = -76$$
$$x_{57} = 92$$
$$x_{58} = 48$$
$$x_{59} = -0.266574298765612$$
$$x_{60} = 83.2522053201295$$
$$x_{61} = -72$$
$$x_{62} = -65.75$$
$$x_{63} = -40$$
$$x_{64} = 68$$
$$x_{65} = -20$$
$$x_{66} = -65.1984042550898$$
$$x_{67} = -2$$
$$x_{68} = -17.2787595947439$$
$$x_{69} = 95.8185759344887$$
$$x_{70} = -52$$
$$x_{71} = -78$$
$$x_{72} = -36.1283155162826$$
$$x_{73} = 1.5707963267949$$
$$x_{74} = 62.6154246585311$$
$$x_{75} = 30$$
$$x_{76} = 62$$
$$x_{77} = -61.261056745001$$
$$x_{78} = -73.8274273593601$$
$$x_{79} = 6$$
$$x_{80} = 64.4026493985908$$
$$x_{81} = -14$$
$$x_{82} = -2350.21477648167$$
$$x_{83} = -48.6946861306418$$
$$x_{84} = -21.75$$
$$x_{85} = 98$$
$$x_{86} = -23.5619449019235$$
$$x_{87} = -42.4115008234622$$
$$x_{88} = 32.9867228626928$$
$$x_{89} = -10.9955742875643$$
$$x_{90} = -80.1106126665397$$
$$x_{91} = 22$$
Signos de extremos en los puntos:
(74, 0)
(-98.96016858807849, 2)
(-4.71238898038469, 2)
(-88, 0)
(-86.39379797371932, 2)
(12, 0)
(94, 0)
(39.269908169872416, 2)
(-34, 0)
(-21.77500195076154, 0)
(-96, 0)
(60, 0)
(-26, 0)
(54, 0)
(10, 0)
(-46, 0)
(-54.977871437821385, 2)
(26.703537555513243, 2)
(-82, 0)
(20.420352248333657, 2)
(-92.6769832808989, 2)
(-64, 0)
(18, 0)
(-28, 0)
(76.96902001294994, 2)
(58.119464091411174, 2)
(-29.845130209103036, 2)
(-278.3820903286233, 0)
(86, 0)
(66, 0)
(4, 0)
(-44, 0)
(-90, 0)
(42, 0)
(-31.985909102429677, 0)
(50, 0)
(7.853981633974483, 2)
(-8, 0)
(89.53539062730911, 2)
(-58, 0)
(36, 0)
(51.83627878423159, 2)
(70.68583470577035, 2)
(-84, 0)
(45.553093477052, 2)
(56, 0)
(16, 0)
(-38, 0)
(-70, 0)
(80, 0)
(14.137166941154069, 2)
(-67.54424205218055, 2)
(24, 0)
(100, 0)
(-32, 0)
(-76, 0)
(92, 0)
(48, 0)
(-0.26657429876561184, 0)
(83.25220532012952, 2)
(-72, 0)
(-65.75, 0)
(-40, 0)
(68, 0)
(-20, 0)
(-65.19840425508981, 0)
(-2, 0)
(-17.278759594743864, 2)
(95.81857593448869, 2)
(-52, 0)
(-78, 0)
(-36.12831551628262, 2)
(1.5707963267948966, 2)
(62.6154246585311, 0)
(30, 0)
(62, 0)
(-61.26105674500097, 2)
(-73.82742735936014, 2)
(6, 0)
(64.40264939859077, 2)
(-14, 0)
(-2350.2147764816746, 0)
(-48.6946861306418, 2)
(-21.75, 0)
(98, 0)
(-23.56194490192345, 2)
(-42.411500823462205, 2)
(32.98672286269283, 2)
(-10.995574287564276, 2)
(-80.11061266653972, 2)
(22, 0)
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:
La función no tiene puntos mínimos
Puntos máximos de la función:
$$x_{91} = -98.9601685880785$$
$$x_{91} = -4.71238898038469$$
$$x_{91} = -86.3937979737193$$
$$x_{91} = 39.2699081698724$$
$$x_{91} = -54.9778714378214$$
$$x_{91} = 26.7035375555132$$
$$x_{91} = 20.4203522483337$$
$$x_{91} = -92.6769832808989$$
$$x_{91} = 76.9690200129499$$
$$x_{91} = 58.1194640914112$$
$$x_{91} = -29.845130209103$$
$$x_{91} = 7.85398163397448$$
$$x_{91} = 89.5353906273091$$
$$x_{91} = 51.8362787842316$$
$$x_{91} = 70.6858347057703$$
$$x_{91} = 45.553093477052$$
$$x_{91} = 14.1371669411541$$
$$x_{91} = -67.5442420521806$$
$$x_{91} = 83.2522053201295$$
$$x_{91} = -17.2787595947439$$
$$x_{91} = 95.8185759344887$$
$$x_{91} = -36.1283155162826$$
$$x_{91} = 1.5707963267949$$
$$x_{91} = -61.261056745001$$
$$x_{91} = -73.8274273593601$$
$$x_{91} = 64.4026493985908$$
$$x_{91} = -48.6946861306418$$
$$x_{91} = -23.5619449019235$$
$$x_{91} = -42.4115008234622$$
$$x_{91} = 32.9867228626928$$
$$x_{91} = -10.9955742875643$$
$$x_{91} = -80.1106126665397$$
Decrece en los intervalos
$$\left(-\infty, -98.9601685880785\right]$$
Crece en los intervalos
$$\left[95.8185759344887, \infty\right)$$
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
$$- \sin{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} - \sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)} \delta\left(\sin{\left(x \right)}\right) = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \sin{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} - \sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)} \delta\left(\sin{\left(x \right)}\right) = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right|\right) = \left\langle -1, 1\right\rangle + \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -1, 1\right\rangle + \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to \infty}\left(\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right|\right) = \left\langle -1, 1\right\rangle + \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -1, 1\right\rangle + \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to -\infty}\left(\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right|\right) = \left\langle -1, 1\right\rangle + \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -1, 1\right\rangle + \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to \infty}\left(\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right|\right) = \left\langle -1, 1\right\rangle + \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -1, 1\right\rangle + \left|{\left\langle -1, 1\right\rangle}\right|$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x) + Abs(sin(x)), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right|}{x}\right) = 0$$
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:
$$\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right| = - \sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right|$$
- No
$$\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right| = \sin{\left(x \right)} - \left|{\sin{\left(x \right)}}\right|$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right| = - \sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right|$$
- No
$$\sin{\left(x \right)} + \left|{\sin{\left(x \right)}}\right| = \sin{\left(x \right)} - \left|{\sin{\left(x \right)}}\right|$$
- No
es decir, función
no es
par ni impar