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$$\frac{\operatorname{sign}{\left(x - \sqrt{2} \right)}}{x - \sqrt{2}} - \frac{\left|{x - \sqrt{2}}\right|}{\left(x - \sqrt{2}\right)^{2}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 94$$
$$x_{2} = 98$$
$$x_{3} = 46$$
$$x_{4} = -57.75$$
$$x_{5} = 24$$
$$x_{6} = -24$$
$$x_{7} = 22$$
$$x_{8} = -42$$
$$x_{9} = -74$$
$$x_{10} = 64$$
$$x_{11} = 36$$
$$x_{12} = -8$$
$$x_{13} = -26$$
$$x_{14} = -2$$
$$x_{15} = 86$$
$$x_{16} = 66.125$$
$$x_{17} = 42$$
$$x_{18} = -81.75$$
$$x_{19} = -36$$
$$x_{20} = 100$$
$$x_{21} = 12$$
$$x_{22} = 10.25$$
$$x_{23} = 90$$
$$x_{24} = -88$$
$$x_{25} = -49.75$$
$$x_{26} = -92$$
$$x_{27} = 52$$
$$x_{28} = -44$$
$$x_{29} = -47.75$$
$$x_{30} = 6$$
$$x_{31} = 38$$
$$x_{32} = -59.75$$
$$x_{33} = 96$$
$$x_{34} = -84$$
$$x_{35} = 40.25$$
$$x_{36} = -98$$
$$x_{37} = 62$$
$$x_{38} = -28$$
$$x_{39} = -17.75$$
$$x_{40} = 14$$
$$x_{41} = 30$$
$$x_{42} = -14$$
$$x_{43} = -72$$
$$x_{44} = 0$$
$$x_{45} = -20$$
$$x_{46} = -52$$
$$x_{47} = 68$$
$$x_{48} = 32$$
$$x_{49} = 28$$
$$x_{50} = 76.25$$
$$x_{51} = -16$$
$$x_{52} = 44$$
$$x_{53} = -6$$
$$x_{54} = 60.25$$
$$x_{55} = 16$$
$$x_{56} = -78$$
$$x_{57} = -65.75$$
$$x_{58} = 4$$
$$x_{59} = -56$$
$$x_{60} = -10$$
$$x_{61} = 74$$
$$x_{62} = -34$$
$$x_{63} = 18.25$$
$$x_{64} = 56.25$$
$$x_{65} = 84$$
$$x_{66} = -11.75$$
$$x_{67} = -68$$
$$x_{68} = 50.25$$
$$x_{69} = -94$$
$$x_{70} = -86$$
$$x_{71} = -4$$
$$x_{72} = -79.75$$
$$x_{73} = -96$$
$$x_{74} = -70$$
$$x_{75} = 54.25$$
$$x_{76} = 8$$
$$x_{77} = 82$$
$$x_{78} = -39.75$$
$$x_{79} = -76$$
$$x_{80} = -38$$
$$x_{81} = -32$$
$$x_{82} = -30$$
$$x_{83} = -45.75$$
$$x_{84} = 20$$
$$x_{85} = 58$$
$$x_{86} = 78$$
$$x_{87} = 70$$
$$x_{88} = -53.75$$
$$x_{89} = 92$$
$$x_{90} = -90$$
$$x_{91} = 72$$
$$x_{92} = 88$$
$$x_{93} = 2$$
$$x_{94} = -64$$
$$x_{95} = -100$$
$$x_{96} = -22$$
$$x_{97} = 34$$
$$x_{98} = 26$$
Signos de extremos en los puntos:
(94, 1)
(98, 1)
(46, 1)
___
57.75 + \/ 2
(-57.75, --------------)
___
-57.75 - \/ 2
(24, 1)
___
24 + \/ 2
(-24, -------------)
___
-24 - \/ 2
(22, 1)
___
42 + \/ 2
(-42, -------------)
___
-42 - \/ 2
___
74 + \/ 2
(-74, -------------)
___
-74 - \/ 2
(64, 1)
(36, 1)
___
8 + \/ 2
(-8, ------------)
___
-8 - \/ 2
___
26 + \/ 2
(-26, -------------)
___
-26 - \/ 2
___
2 + \/ 2
(-2, ------------)
___
-2 - \/ 2
(86, 1)
(66.125, 1)
(42, 1)
___
81.75 + \/ 2
(-81.75, --------------)
___
-81.75 - \/ 2
___
36 + \/ 2
(-36, -------------)
___
-36 - \/ 2
(100, 1)
(12, 1)
(10.25, 1)
(90, 1)
___
88 + \/ 2
(-88, -------------)
___
-88 - \/ 2
___
49.75 + \/ 2
(-49.75, --------------)
___
-49.75 - \/ 2
___
92 + \/ 2
(-92, -------------)
___
-92 - \/ 2
(52, 1)
___
44 + \/ 2
(-44, -------------)
___
-44 - \/ 2
___
47.75 + \/ 2
(-47.75, --------------)
___
-47.75 - \/ 2
(6, 1)
(38, 1)
___
59.75 + \/ 2
(-59.75, --------------)
___
-59.75 - \/ 2
(96, 1)
___
84 + \/ 2
(-84, -------------)
___
-84 - \/ 2
(40.25, 1)
___
98 + \/ 2
(-98, -------------)
___
-98 - \/ 2
(62, 1)
___
28 + \/ 2
(-28, -------------)
___
-28 - \/ 2
___
17.75 + \/ 2
(-17.75, --------------)
___
-17.75 - \/ 2
(14, 1)
(30, 1)
___
14 + \/ 2
(-14, -------------)
___
-14 - \/ 2
___
72 + \/ 2
(-72, -------------)
___
-72 - \/ 2
(0, -1)
___
20 + \/ 2
(-20, -------------)
___
-20 - \/ 2
___
52 + \/ 2
(-52, -------------)
___
-52 - \/ 2
(68, 1)
(32, 1)
(28, 1)
(76.25, 1)
___
16 + \/ 2
(-16, -------------)
___
-16 - \/ 2
(44, 1)
___
6 + \/ 2
(-6, ------------)
___
-6 - \/ 2
(60.25, 1)
(16, 1)
___
78 + \/ 2
(-78, -------------)
___
-78 - \/ 2
___
65.75 + \/ 2
(-65.75, --------------)
___
-65.75 - \/ 2
(4, 1)
___
56 + \/ 2
(-56, -------------)
___
-56 - \/ 2
___
10 + \/ 2
(-10, -------------)
___
-10 - \/ 2
(74, 1)
___
34 + \/ 2
(-34, -------------)
___
-34 - \/ 2
(18.25, 1)
(56.25, 1)
(84, 1)
___
11.75 + \/ 2
(-11.75, --------------)
___
-11.75 - \/ 2
___
68 + \/ 2
(-68, -------------)
___
-68 - \/ 2
(50.25, 1)
___
94 + \/ 2
(-94, -------------)
___
-94 - \/ 2
___
86 + \/ 2
(-86, -------------)
___
-86 - \/ 2
___
4 + \/ 2
(-4, ------------)
___
-4 - \/ 2
___
79.75 + \/ 2
(-79.75, --------------)
___
-79.75 - \/ 2
___
96 + \/ 2
(-96, -------------)
___
-96 - \/ 2
___
70 + \/ 2
(-70, -------------)
___
-70 - \/ 2
(54.25, 1)
(8, 1)
(82, 1)
___
39.75 + \/ 2
(-39.75, --------------)
___
-39.75 - \/ 2
___
76 + \/ 2
(-76, -------------)
___
-76 - \/ 2
___
38 + \/ 2
(-38, -------------)
___
-38 - \/ 2
___
32 + \/ 2
(-32, -------------)
___
-32 - \/ 2
___
30 + \/ 2
(-30, -------------)
___
-30 - \/ 2
___
45.75 + \/ 2
(-45.75, --------------)
___
-45.75 - \/ 2
(20, 1)
(58, 1)
(78, 1)
(70, 1)
___
53.75 + \/ 2
(-53.75, --------------)
___
-53.75 - \/ 2
(92, 1)
___
90 + \/ 2
(-90, -------------)
___
-90 - \/ 2
(72, 1)
(88, 1)
(2, 1)
___
64 + \/ 2
(-64, -------------)
___
-64 - \/ 2
___
100 + \/ 2
(-100, --------------)
___
-100 - \/ 2
___
22 + \/ 2
(-22, -------------)
___
-22 - \/ 2
(34, 1)
(26, 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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
No cambia el valor en todo el eje numérico