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$$\frac{\sin{\left(x \right)} \cos{\left(\left|{x}\right| \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin{\left(\left|{x}\right| \right)} \operatorname{sign}{\left(x \right)}}{\cos{\left(x \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -34$$
$$x_{2} = 8$$
$$x_{3} = -6$$
$$x_{4} = -41.75$$
$$x_{5} = -84$$
$$x_{6} = 85.4791666666667$$
$$x_{7} = -74$$
$$x_{8} = -10$$
$$x_{9} = 24$$
$$x_{10} = 72$$
$$x_{11} = -44$$
$$x_{12} = -94$$
$$x_{13} = -12.25$$
$$x_{14} = 62$$
$$x_{15} = -59.75$$
$$x_{16} = -26$$
$$x_{17} = 50$$
$$x_{18} = 14$$
$$x_{19} = 20$$
$$x_{20} = -52$$
$$x_{21} = -54$$
$$x_{22} = 18$$
$$x_{23} = 32$$
$$x_{24} = -80$$
$$x_{25} = 46.1666666666667$$
$$x_{26} = -20$$
$$x_{27} = -28$$
$$x_{28} = -64$$
$$x_{29} = 78$$
$$x_{30} = -4$$
$$x_{31} = 16$$
$$x_{32} = 34$$
$$x_{33} = 58$$
$$x_{34} = 42.25$$
$$x_{35} = -90$$
$$x_{36} = 76.25$$
$$x_{37} = 74$$
$$x_{38} = -100$$
$$x_{39} = -8$$
$$x_{40} = 68$$
$$x_{41} = 2$$
$$x_{42} = -56$$
$$x_{43} = 88$$
$$x_{44} = -22$$
$$x_{45} = 70$$
$$x_{46} = 28$$
$$x_{47} = -88$$
$$x_{48} = 12.25$$
$$x_{49} = 54$$
$$x_{50} = -78$$
$$x_{51} = 30.25$$
$$x_{52} = 38.125$$
$$x_{53} = 52$$
$$x_{54} = 84$$
$$x_{55} = 10$$
$$x_{56} = -98$$
$$x_{57} = 6$$
$$x_{58} = 80$$
$$x_{59} = 60.25$$
$$x_{60} = -62$$
$$x_{61} = 56$$
$$x_{62} = -32$$
$$x_{63} = 44$$
$$x_{64} = 96.7083333333333$$
$$x_{65} = 82$$
$$x_{66} = -68$$
$$x_{67} = -58$$
$$x_{68} = -24$$
$$x_{69} = -85.25$$
$$x_{70} = -37.7777777777778$$
$$x_{71} = -35.8333333333333$$
$$x_{72} = 64$$
$$x_{73} = 22$$
$$x_{74} = 92$$
$$x_{75} = 100$$
$$x_{76} = -70$$
$$x_{77} = -92$$
$$x_{78} = -72$$
$$x_{79} = -75.75$$
$$x_{80} = 0$$
$$x_{81} = 26$$
$$x_{82} = 36.25$$
$$x_{83} = -82$$
$$x_{84} = -50$$
$$x_{85} = -45.75$$
$$x_{86} = 40.25$$
$$x_{87} = 90$$
$$x_{88} = -47.75$$
$$x_{89} = -16$$
$$x_{90} = 4$$
$$x_{91} = -39.75$$
$$x_{92} = -14$$
$$x_{93} = 94$$
$$x_{94} = 98$$
$$x_{95} = -95.75$$
$$x_{96} = -29.75$$
$$x_{97} = -66$$
$$x_{98} = -18$$
$$x_{99} = 66$$
$$x_{100} = 48.25$$
$$x_{101} = -2$$
Signos de extremos en los puntos:
(-34, 1)
(8, 1)
(-6, 1)
(-41.75, 1)
(-84, 1)
(85.47916666666667, 1)
(-74, 1)
(-10, 1)
(24, 1)
(72, 1)
(-44, 1)
(-94, 1)
(-12.25, 1)
(62, 1)
(-59.75, 1)
(-26, 1)
(50, 1)
(14, 1)
(20, 1)
(-52, 1)
(-54, 1)
(18, 1)
(32, 1)
(-80, 1)
(46.166666666666664, 1)
(-20, 1)
(-28, 1)
(-64, 1)
(78, 1)
(-4, 1)
(16, 1)
(34, 1)
(58, 1)
(42.25, 1)
(-90, 1)
(76.25, 1)
(74, 1)
(-100, 1)
(-8, 1)
(68, 1)
(2, 1)
(-56, 1)
(88, 1)
(-22, 1)
(70, 1)
(28, 1)
(-88, 1)
(12.25, 1)
(54, 1)
(-78, 1)
(30.25, 1)
(38.125, 1)
(52, 1)
(84, 1)
(10, 1)
(-98, 1)
(6, 1)
(80, 1)
(60.25, 1)
(-62, 1)
(56, 1)
(-32, 1)
(44, 1)
(96.70833333333333, 1)
(82, 1)
(-68, 1)
(-58, 1)
(-24, 1)
(-85.25, 1)
(-37.77777777777778, 1)
(-35.833333333333336, 1)
(64, 1)
(22, 1)
(92, 1)
(100, 1)
(-70, 1)
(-92, 1)
(-72, 1)
(-75.75, 1)
(0, 1)
(26, 1)
(36.25, 1)
(-82, 1)
(-50, 1)
(-45.75, 1)
(40.25, 1)
(90, 1)
(-47.75, 1)
(-16, 1)
(4, 1)
(-39.75, 1)
(-14, 1)
(94, 1)
(98, 1)
(-95.75, 1)
(-29.75, 1)
(-66, 1)
(-18, 1)
(66, 1)
(48.25, 1)
(-2, 1)
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:
$$x_{1} = -10$$
$$x_{2} = 72$$
$$x_{3} = -94$$
$$x_{4} = 58$$
$$x_{5} = -90$$
$$x_{6} = 10$$
$$x_{7} = -58$$
$$x_{8} = -72$$
$$x_{9} = 90$$
$$x_{10} = 94$$
Puntos máximos de la función:
$$x_{10} = -28$$
$$x_{10} = 28$$
Decrece en los intervalos
$$\left[94, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -94\right]$$