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$$\frac{\left(- \tan^{2}{\left(x \right)} - 1\right) \tan{\left(\left|{x}\right| \right)}}{\tan^{2}{\left(x \right)}} + \frac{\left(\tan^{2}{\left(\left|{x}\right| \right)} + 1\right) \operatorname{sign}{\left(x \right)}}{\tan{\left(x \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 90$$
$$x_{2} = -33.75$$
$$x_{3} = 18$$
$$x_{4} = 8.25$$
$$x_{5} = -85.75$$
$$x_{6} = 30$$
$$x_{7} = -4$$
$$x_{8} = 94$$
$$x_{9} = -100$$
$$x_{10} = -96.15$$
$$x_{11} = -21.7647058823529$$
$$x_{12} = -6$$
$$x_{13} = -69.75$$
$$x_{14} = 38$$
$$x_{15} = 46$$
$$x_{16} = 2$$
$$x_{17} = 6$$
$$x_{18} = 66$$
$$x_{19} = 22.25$$
$$x_{20} = 82$$
$$x_{21} = -60$$
$$x_{22} = -90$$
$$x_{23} = 52.1666666666667$$
$$x_{24} = -88$$
$$x_{25} = -98$$
$$x_{26} = -44$$
$$x_{27} = -74$$
$$x_{28} = 72$$
$$x_{29} = 10$$
$$x_{30} = 48$$
$$x_{31} = -79.75$$
$$x_{32} = -14$$
$$x_{33} = -53.75$$
$$x_{34} = -62$$
$$x_{35} = -46$$
$$x_{36} = 50.25$$
$$x_{37} = 42$$
$$x_{38} = 40.125$$
$$x_{39} = -16$$
$$x_{40} = -42$$
$$x_{41} = 64$$
$$x_{42} = 84.125$$
$$x_{43} = -35.8333333333333$$
$$x_{44} = 12$$
$$x_{45} = 4$$
$$x_{46} = 78.25$$
$$x_{47} = 16$$
$$x_{48} = 92$$
$$x_{49} = -68$$
$$x_{50} = 74$$
$$x_{51} = -83.875$$
$$x_{52} = 54.25$$
$$x_{53} = -77.75$$
$$x_{54} = -76$$
$$x_{55} = -82$$
$$x_{56} = 76$$
$$x_{57} = 24.25$$
$$x_{58} = -26$$
$$x_{59} = -52.2$$
$$x_{60} = -38$$
$$x_{61} = -58$$
$$x_{62} = 34.125$$
$$x_{63} = -7.75$$
$$x_{64} = 26$$
$$x_{65} = -48$$
$$x_{66} = -18$$
$$x_{67} = -27.875$$
$$x_{68} = 60$$
$$x_{69} = 100$$
$$x_{70} = 88$$
$$x_{71} = -10$$
$$x_{72} = -20$$
$$x_{73} = -66$$
$$x_{74} = -94$$
$$x_{75} = -12$$
$$x_{76} = -72$$
$$x_{77} = 80.25$$
$$x_{78} = -30$$
$$x_{79} = 20$$
$$x_{80} = 14$$
$$x_{81} = -55.75$$
$$x_{82} = 36.25$$
$$x_{83} = 32$$
$$x_{84} = -32$$
$$x_{85} = 44$$
$$x_{86} = 62$$
$$x_{87} = -49.75$$
$$x_{88} = -64$$
$$x_{89} = 28.25$$
$$x_{90} = 68$$
$$x_{91} = -24.25$$
$$x_{92} = 58$$
$$x_{93} = 96.1666666666667$$
$$x_{94} = -39.75$$
$$x_{95} = 86.1388888888889$$
$$x_{96} = -92$$
$$x_{97} = 98$$
$$x_{98} = 56.125$$
$$x_{99} = -2$$
Signos de extremos en los puntos:
(90, 1)
(-33.75, -1)
(18, 1)
(8.25, 1)
(-85.75, -1)
(30, 1)
(-4, -1)
(94, 1)
(-100, -1)
(-96.15, -1)
(-21.764705882352942, -1)
(-6, -1)
(-69.75, -1)
(38, 1)
(46, 1)
(2, 1)
(6, 1)
(66, 1)
(22.25, 1)
(82, 1)
(-60, -1)
(-90, -1)
(52.166666666666664, 1)
(-88, -1)
(-98, -1)
(-44, -1)
(-74, -1)
(72, 1)
(10, 1)
(48, 1)
(-79.75, -1)
(-14, -1)
(-53.75, -1)
(-62, -1)
(-46, -1)
(50.25, 1)
(42, 1)
(40.125, 1)
(-16, -1)
(-42, -1)
(64, 1)
(84.125, 1)
(-35.833333333333336, -1)
(12, 1)
(4, 1)
(78.25, 1)
(16, 1)
(92, 1)
(-68, -1)
(74, 1)
(-83.875, -1)
(54.25, 1)
(-77.75, -1)
(-76, -1)
(-82, -1)
(76, 1)
(24.25, 1)
(-26, -1)
(-52.2, -1)
(-38, -1)
(-58, -1)
(34.125, 1)
(-7.75, -1)
(26, 1)
(-48, -1)
(-18, -1)
(-27.875, -1)
(60, 1)
(100, 1)
(88, 1)
(-10, -1)
(-20, -1)
(-66, -1)
(-94, -1)
(-12, -1)
(-72, -1)
(80.25, 1)
(-30, -1)
(20, 1)
(14, 1)
(-55.75, -1)
(36.25, 1)
(32, 1)
(-32, -1)
(44, 1)
(62, 1)
(-49.75, -1)
(-64, -1)
(28.25, 1)
(68, 1)
(-24.25, -1)
(58, 1)
(96.16666666666667, 1)
(-39.75, -1)
(86.13888888888889, 1)
(-92, -1)
(98, 1)
(56.125, 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:
$$x_{1} = 48$$
$$x_{2} = 40.125$$
$$x_{3} = 74$$
$$x_{4} = 54.25$$
$$x_{5} = 76$$
$$x_{6} = 80.25$$
Puntos máximos de la función:
$$x_{6} = -74$$
$$x_{6} = -53.75$$
$$x_{6} = -76$$
$$x_{6} = -48$$
$$x_{6} = -49.75$$
Decrece en los intervalos
$$\left[80.25, \infty\right)$$
Crece en los intervalos
$$\left[-48, 40.125\right]$$