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$$\frac{\cos{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)}} - \frac{\cos{\left(x \right)} \left|{\sin{\left(x \right)}}\right|}{\sin^{2}{\left(x \right)}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 74$$
$$x_{2} = 84$$
$$x_{3} = -88$$
$$x_{4} = 8.25$$
$$x_{5} = -62$$
$$x_{6} = 28$$
$$x_{7} = -48$$
$$x_{8} = 12$$
$$x_{9} = 94$$
$$x_{10} = -7.75$$
$$x_{11} = 26$$
$$x_{12} = 64$$
$$x_{13} = 76$$
$$x_{14} = 42.25$$
$$x_{15} = -56$$
$$x_{16} = 24.25$$
$$x_{17} = 5.96428571428571$$
$$x_{18} = -96$$
$$x_{19} = -26$$
$$x_{20} = 54$$
$$x_{21} = 10$$
$$x_{22} = 14$$
$$x_{23} = -91.75$$
$$x_{24} = -45.7$$
$$x_{25} = -66$$
$$x_{26} = 82.25$$
$$x_{27} = -74$$
$$x_{28} = -64$$
$$x_{29} = 46.25$$
$$x_{30} = -36$$
$$x_{31} = 18$$
$$x_{32} = -28$$
$$x_{33} = -18$$
$$x_{34} = -77.75$$
$$x_{35} = -37.75$$
$$x_{36} = 66$$
$$x_{37} = 4$$
$$x_{38} = 60.25$$
$$x_{39} = -50$$
$$x_{40} = 88$$
$$x_{41} = -90$$
$$x_{42} = -12$$
$$x_{43} = 32$$
$$x_{44} = -59.75$$
$$x_{45} = 50$$
$$x_{46} = -68$$
$$x_{47} = -58$$
$$x_{48} = -23.4833333333333$$
$$x_{49} = -99.75$$
$$x_{50} = 36$$
$$x_{51} = 92.1666666666667$$
$$x_{52} = -84$$
$$x_{53} = 90$$
$$x_{54} = -54$$
$$x_{55} = 20$$
$$x_{56} = 56$$
$$x_{57} = -5.5$$
$$x_{58} = 2$$
$$x_{59} = 80$$
$$x_{60} = -70$$
$$x_{61} = -85.75$$
$$x_{62} = 100.25$$
$$x_{63} = 34.0833333333333$$
$$x_{64} = 52$$
$$x_{65} = 22.2424242424242$$
$$x_{66} = -81.95$$
$$x_{67} = -15.75$$
$$x_{68} = 78.25$$
$$x_{69} = 40$$
$$x_{70} = -32$$
$$x_{71} = -76$$
$$x_{72} = -94$$
$$x_{73} = 48$$
$$x_{74} = -30$$
$$x_{75} = 86.5$$
$$x_{76} = -72$$
$$x_{77} = -40$$
$$x_{78} = 68$$
$$x_{79} = -20$$
$$x_{80} = -2$$
$$x_{81} = -33.5$$
$$x_{82} = 72$$
$$x_{83} = 44.25$$
$$x_{84} = -52$$
$$x_{85} = 58$$
$$x_{86} = -4$$
$$x_{87} = -98$$
$$x_{88} = 30$$
$$x_{89} = 62$$
$$x_{90} = 16.1666666666667$$
$$x_{91} = -80$$
$$x_{92} = 38.25$$
$$x_{93} = -14$$
$$x_{94} = -43.6964285714286$$
$$x_{95} = -42.25$$
$$x_{96} = -21.75$$
$$x_{97} = 70$$
$$x_{98} = 96$$
$$x_{99} = 98$$
$$x_{100} = -10$$
Signos de extremos en los puntos:
(74, 1.11022302462516e-16)
(84, 2)
(-88, 0)
(8.25, 2)
(-62, 2)
(28, 2)
(-48, 2)
(12, 0)
(94, 1.11022302462516e-16)
(-7.75, 1.11022302462516e-16)
(26, 2)
(64, 2)
(76, 2)
(42.25, 1.11022302462516e-16)
(-56, 2)
(24.25, 0)
(5.964285714285714, 0)
(-96, 1.11022302462516e-16)
(-26, 0)
(54, 0)
(10, 0)
(14, 2)
(-91.75, 2)
(-45.7, 1.11022302462516e-16)
(-66, 2)
(82.25, 2)
(-74, 2)
(-64, 0)
(46.25, 2)
(-36, 2)
(18, 0)
(-28, 0)
(-18, 2)
(-77.75, 0)
(-37.75, 0)
(66, 0)
(4, 1.11022302462516e-16)
(60.25, 0)
(-50, 2)
(88, 2)
(-90, 0)
(-12, 2)
(32, 2)
(-59.75, 2)
(50, 0)
(-68, 2)
(-58, 1.11022302462516e-16)
(-23.483333333333334, 2)
(-99.75, 2)
(36, 0)
(92.16666666666667, 0)
(-84, 1.11022302462516e-16)
(90, 2)
(-54, 2)
(20, 2)
(56, 0)
(-5.5, 2)
(2, 2)
(80, 0)
(-70, 0)
(-85.75, 2)
(100.25, 0)
(34.083333333333336, 2)
(52, 2)
(22.242424242424242, 0)
(-81.95, 0)
(-15.75, 2)
(78.25, 2)
(40, 2)
(-32, 1.11022302462516e-16)
(-76, 0)
(-94, 2)
(48, 0)
(-30, 2)
(86.5, 0)
(-72, 0)
(-40, 1.11022302462516e-16)
(68, 1.11022302462516e-16)
(-20, 0)
(-2, 0)
(-33.5, 0)
(72, 2)
(44.25, 2)
(-52, 0)
(58, 2)
(-4, 2)
(-98, 2)
(30, 0)
(62, 1.11022302462516e-16)
(16.166666666666668, 0)
(-80, 2)
(38.25, 2)
(-14, 0)
(-43.69642857142857, 2)
(-42.25, 2)
(-21.75, 0)
(70, 2)
(96, 2)
(98, 0)
(-10, 2)
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:
$$x_{1} = 36$$
$$x_{2} = -70$$
$$x_{3} = -33.5$$
Puntos máximos de la función:
$$x_{3} = 74$$
$$x_{3} = 94$$
$$x_{3} = -7.75$$
$$x_{3} = -96$$
$$x_{3} = 4$$
$$x_{3} = -58$$
$$x_{3} = -32$$
Decrece en los intervalos
$$\left[36, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -70\right]$$