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$$\frac{\operatorname{sign}{\left(x - 3 \right)}}{x - 3} - \frac{\left|{x - 3}\right|}{\left(x - 3\right)^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -16$$
$$x_{2} = -46$$
$$x_{3} = 50$$
$$x_{4} = -84$$
$$x_{5} = -68$$
$$x_{6} = -90$$
$$x_{7} = -98$$
$$x_{8} = -38$$
$$x_{9} = -14$$
$$x_{10} = 52$$
$$x_{11} = 72$$
$$x_{12} = 76$$
$$x_{13} = 74$$
$$x_{14} = -22$$
$$x_{15} = -10$$
$$x_{16} = -48$$
$$x_{17} = 48$$
$$x_{18} = -8$$
$$x_{19} = -42$$
$$x_{20} = 0$$
$$x_{21} = 82$$
$$x_{22} = -58$$
$$x_{23} = 70$$
$$x_{24} = -28$$
$$x_{25} = -60$$
$$x_{26} = 34$$
$$x_{27} = -92$$
$$x_{28} = -86$$
$$x_{29} = 16$$
$$x_{30} = 90$$
$$x_{31} = 78$$
$$x_{32} = -100$$
$$x_{33} = -40$$
$$x_{34} = 66$$
$$x_{35} = 20$$
$$x_{36} = -76$$
$$x_{37} = 30$$
$$x_{38} = -34$$
$$x_{39} = 88$$
$$x_{40} = -6$$
$$x_{41} = 6$$
$$x_{42} = 38$$
$$x_{43} = 2$$
$$x_{44} = -78$$
$$x_{45} = 36$$
$$x_{46} = 8$$
$$x_{47} = 32$$
$$x_{48} = -56$$
$$x_{49} = 42$$
$$x_{50} = -66$$
$$x_{51} = 96$$
$$x_{52} = -82$$
$$x_{53} = 86$$
$$x_{54} = -30$$
$$x_{55} = -88$$
$$x_{56} = -12$$
$$x_{57} = 44$$
$$x_{58} = 94$$
$$x_{59} = -72$$
$$x_{60} = 56$$
$$x_{61} = 12$$
$$x_{62} = 68$$
$$x_{63} = 100$$
$$x_{64} = 24$$
$$x_{65} = 4$$
$$x_{66} = 92$$
$$x_{67} = -74$$
$$x_{68} = -20$$
$$x_{69} = -36$$
$$x_{70} = -32$$
$$x_{71} = 62$$
$$x_{72} = 14$$
$$x_{73} = -26$$
$$x_{74} = 18$$
$$x_{75} = 40$$
$$x_{76} = -62$$
$$x_{77} = 98$$
$$x_{78} = 10$$
$$x_{79} = 54$$
$$x_{80} = 64$$
$$x_{81} = 84$$
$$x_{82} = 46$$
$$x_{83} = -2$$
$$x_{84} = -44$$
$$x_{85} = 28$$
$$x_{86} = 26$$
$$x_{87} = -4$$
$$x_{88} = 58$$
$$x_{89} = -54$$
$$x_{90} = -64$$
$$x_{91} = -70$$
$$x_{92} = -96$$
$$x_{93} = -52$$
$$x_{94} = 22$$
$$x_{95} = -80$$
$$x_{96} = 60$$
$$x_{97} = -24$$
$$x_{98} = -18$$
$$x_{99} = 80$$
$$x_{100} = -94$$
$$x_{101} = -50$$
Signos de extremos en los puntos:
(-16, -2)
(-46, -2)
(50, 0)
(-84, -2)
(-68, -2)
(-90, -2)
(-98, -2)
(-38, -2)
(-14, -2)
(52, -1.11022302462516e-16)
(72, 0)
(76, 0)
(74, 0)
(-22, -2)
(-10, -2)
(-48, -2)
(48, 0)
(-8, -2)
(-42, -2)
(0, -2)
(82, 0)
(-58, -2)
(70, 0)
(-28, -2)
(-60, -2)
(34, 0)
(-92, -2)
(-86, -2)
(16, 0)
(90, 0)
(78, 0)
(-100, -2)
(-40, -2)
(66, 0)
(20, 0)
(-76, -2)
(30, 0)
(-34, -2)
(88, 0)
(-6, -2)
(6, 0)
(38, 0)
(2, -2)
(-78, -2)
(36, 0)
(8, 0)
(32, 0)
(-56, -2)
(42, 0)
(-66, -2)
(96, 0)
(-82, -2)
(86, 0)
(-30, -2)
(-88, -2)
(-12, -2)
(44, 0)
(94, 0)
(-72, -2)
(56, 0)
(12, 0)
(68, 0)
(100, 0)
(24, 0)
(4, 0)
(92, 0)
(-74, -2)
(-20, -2)
(-36, -2)
(-32, -2)
(62, 0)
(14, 0)
(-26, -2)
(18, 0)
(40, 0)
(-62, -2)
(98, 0)
(10, 0)
(54, 0)
(64, 0)
(84, 0)
(46, 0)
(-2, -2)
(-44, -2)
(28, 0)
(26, 0)
(-4, -2)
(58, 0)
(-54, -2)
(-64, -2)
(-70, -2)
(-96, -2)
(-52, -2)
(22, 0)
(-80, -2)
(60, 0)
(-24, -2)
(-18, -2)
(80, 0)
(-94, -2)
(-50, -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:
$$x_{1} = 52$$
Puntos máximos de la función:
$$x_{1} = 90$$
$$x_{1} = 66$$
Decrece en los intervalos
$$\left[52, 66\right]$$
Crece en los intervalos
$$\left(-\infty, 52\right] \cup \left[90, \infty\right)$$