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$$\frac{\operatorname{sign}{\left(x - 2 \right)}}{x - 2} - \frac{\left|{x - 2}\right|}{\left(x - 2\right)^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 74$$
$$x_{2} = 84$$
$$x_{3} = -88$$
$$x_{4} = 82$$
$$x_{5} = -62$$
$$x_{6} = -100$$
$$x_{7} = 28$$
$$x_{8} = -48$$
$$x_{9} = 12$$
$$x_{10} = 94$$
$$x_{11} = 26$$
$$x_{12} = 64$$
$$x_{13} = -34$$
$$x_{14} = 76$$
$$x_{15} = 44$$
$$x_{16} = -56$$
$$x_{17} = -96$$
$$x_{18} = 78$$
$$x_{19} = 60$$
$$x_{20} = -24$$
$$x_{21} = -26$$
$$x_{22} = 54$$
$$x_{23} = 10$$
$$x_{24} = -42$$
$$x_{25} = -46$$
$$x_{26} = 14$$
$$x_{27} = -82$$
$$x_{28} = -66$$
$$x_{29} = -74$$
$$x_{30} = -64$$
$$x_{31} = -36$$
$$x_{32} = 18$$
$$x_{33} = -28$$
$$x_{34} = -18$$
$$x_{35} = 86$$
$$x_{36} = 66$$
$$x_{37} = 4$$
$$x_{38} = -50$$
$$x_{39} = -44$$
$$x_{40} = 88$$
$$x_{41} = -90$$
$$x_{42} = -12$$
$$x_{43} = 32$$
$$x_{44} = 42$$
$$x_{45} = 50$$
$$x_{46} = -68$$
$$x_{47} = -8$$
$$x_{48} = 0$$
$$x_{49} = -58$$
$$x_{50} = -60$$
$$x_{51} = 36$$
$$x_{52} = -84$$
$$x_{53} = 90$$
$$x_{54} = -54$$
$$x_{55} = 20$$
$$x_{56} = 56$$
$$x_{57} = 46$$
$$x_{58} = 16$$
$$x_{59} = -38$$
$$x_{60} = -70$$
$$x_{61} = 80$$
$$x_{62} = 24$$
$$x_{63} = 52$$
$$x_{64} = 100$$
$$x_{65} = 40$$
$$x_{66} = -32$$
$$x_{67} = -76$$
$$x_{68} = -94$$
$$x_{69} = 92$$
$$x_{70} = 48$$
$$x_{71} = -30$$
$$x_{72} = -72$$
$$x_{73} = -40$$
$$x_{74} = 68$$
$$x_{75} = -20$$
$$x_{76} = -2$$
$$x_{77} = 34$$
$$x_{78} = 72$$
$$x_{79} = -52$$
$$x_{80} = -78$$
$$x_{81} = 58$$
$$x_{82} = -16$$
$$x_{83} = -92$$
$$x_{84} = -4$$
$$x_{85} = -98$$
$$x_{86} = 38$$
$$x_{87} = 30$$
$$x_{88} = -22$$
$$x_{89} = 62$$
$$x_{90} = -86$$
$$x_{91} = -80$$
$$x_{92} = 6$$
$$x_{93} = -14$$
$$x_{94} = 70$$
$$x_{95} = 96$$
$$x_{96} = 98$$
$$x_{97} = -6$$
$$x_{98} = -10$$
$$x_{99} = 22$$
$$x_{100} = 8$$
Signos de extremos en los puntos:
(74, 1)
(84, 1)
(-88, -1)
(82, 1)
(-62, -1)
(-100, -1)
(28, 1)
(-48, -1)
(12, 1)
(94, 1)
(26, 1)
(64, 1)
(-34, -1)
(76, 1)
(44, 1)
(-56, -1)
(-96, -1)
(78, 1)
(60, 1)
(-24, -1)
(-26, -1)
(54, 1)
(10, 1)
(-42, -1)
(-46, -1)
(14, 1)
(-82, -1)
(-66, -1)
(-74, -1)
(-64, -1)
(-36, -1)
(18, 1)
(-28, -1)
(-18, -1)
(86, 1)
(66, 1)
(4, 1)
(-50, -1)
(-44, -1)
(88, 1)
(-90, -1)
(-12, -1)
(32, 1)
(42, 1)
(50, 1)
(-68, -1)
(-8, -1)
(0, -1)
(-58, -1)
(-60, -1)
(36, 1)
(-84, -1)
(90, 1)
(-54, -1)
(20, 1)
(56, 1)
(46, 1)
(16, 1)
(-38, -1)
(-70, -1)
(80, 1)
(24, 1)
(52, 1)
(100, 1)
(40, 1)
(-32, -1)
(-76, -1)
(-94, -1)
(92, 1)
(48, 1)
(-30, -1)
(-72, -1)
(-40, -1)
(68, 1)
(-20, -1)
(-2, -1)
(34, 1)
(72, 1)
(-52, -1)
(-78, -1)
(58, 1)
(-16, -1)
(-92, -1)
(-4, -1)
(-98, -1)
(38, 1)
(30, 1)
(-22, -1)
(62, 1)
(-86, -1)
(-80, -1)
(6, 1)
(-14, -1)
(70, 1)
(96, 1)
(98, 1)
(-6, -1)
(-10, -1)
(22, 1)
(8, 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:
$$x_{1} = -28$$
La función no tiene puntos máximos
Decrece en los intervalos
$$\left[-28, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -28\right]$$