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$$\frac{\left(- \frac{\pi \left(\operatorname{re}{\left(\left(\frac{x}{\operatorname{sign}{\left(x + 2 \right)}} + \frac{2}{\operatorname{sign}{\left(x + 2 \right)}}\right)^{- \pi}\right)} - 3\right) \operatorname{re}{\left(\left(\frac{x}{\operatorname{sign}{\left(x + 2 \right)}} + \frac{2}{\operatorname{sign}{\left(x + 2 \right)}}\right)^{- \pi}\right)} \operatorname{sign}^{2}{\left(x + 2 \right)}}{x + 2} - \frac{\pi \left(\operatorname{im}{\left(\left(\frac{x}{\operatorname{sign}{\left(x + 2 \right)}} + \frac{2}{\operatorname{sign}{\left(x + 2 \right)}}\right)^{- \pi}\right)}\right)^{2} \operatorname{sign}^{2}{\left(x + 2 \right)}}{x + 2}\right) \operatorname{sign}{\left(3 - \left(\frac{x + 2}{\operatorname{sign}{\left(x + 2 \right)}}\right)^{- \pi} \right)}}{3 - \left(\frac{x + 2}{\operatorname{sign}{\left(x + 2 \right)}}\right)^{- \pi}} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 6619.49729962574$$
$$x_{2} = 6241.96367347885$$
$$x_{3} = -3007.70891483234$$
$$x_{4} = 7563.33191389971$$
$$x_{5} = 1522.84580744048$$
$$x_{6} = -8670.70218981557$$
$$x_{7} = 1711.60332158517$$
$$x_{8} = -8104.40097603415$$
$$x_{9} = 7374.56493794176$$
$$x_{10} = -8859.46929267063$$
$$x_{11} = 5864.43020911678$$
$$x_{12} = 2844.17603807134$$
$$x_{13} = 3976.7669250639$$
$$x_{14} = 8695.93419415389$$
$$x_{15} = -7538.09992936435$$
$$x_{16} = 5109.36390908107$$
$$x_{17} = -7915.63394013748$$
$$x_{18} = 2089.12372096819$$
$$x_{19} = 4165.53279983509$$
$$x_{20} = -5839.19827857882$$
$$x_{21} = 8507.16710363322$$
$$x_{22} = 1900.36280352646$$
$$x_{23} = -3762.76946538325$$
$$x_{24} = -5461.66502866337$$
$$x_{25} = 7940.86593222273$$
$$x_{26} = -3574.0039622298$$
$$x_{27} = 6430.7304681089$$
$$x_{28} = 6997.03106138531$$
$$x_{29} = -2818.9445497999$$
$$x_{30} = 1334.09109365846$$
$$x_{31} = 9073.46841902312$$
$$x_{32} = -2441.41724949059$$
$$x_{33} = 1145.3405595432$$
$$x_{34} = -9048.23640967437$$
$$x_{35} = 2655.41202992547$$
$$x_{36} = 5675.66354768655$$
$$x_{37} = -7726.8669240837$$
$$x_{38} = -1497.61577182604$$
$$x_{39} = 4920.59750019097$$
$$x_{40} = 8884.7012995941$$
$$x_{41} = -2063.89272902779$$
$$x_{42} = -6594.26533994979$$
$$x_{43} = -6971.79909055258$$
$$x_{44} = 4731.83117363891$$
$$x_{45} = 3788.00120629721$$
$$x_{46} = -8481.93510205432$$
$$x_{47} = -4140.30100254609$$
$$x_{48} = -6405.49851476393$$
$$x_{49} = -1308.86167948696$$
$$x_{50} = -3951.53515395055$$
$$x_{51} = -3196.47364372814$$
$$x_{52} = 5486.89693990935$$
$$x_{53} = 4354.29881034135$$
$$x_{54} = 8318.40002904729$$
$$x_{55} = 5298.13039151541$$
$$x_{56} = -4517.833099512$$
$$x_{57} = 9451.00269649626$$
$$x_{58} = -8293.16803041807$$
$$x_{59} = 7752.09891253209$$
$$x_{60} = 6808.26416496307$$
$$x_{61} = 6053.19691918408$$
$$x_{62} = 8129.63297150551$$
$$x_{63} = 3410.47034026485$$
$$x_{64} = -9237.00353995899$$
$$x_{65} = -7160.56601068187$$
$$x_{66} = -2252.65455426702$$
$$x_{67} = -1686.37285990474$$
$$x_{68} = 3221.70525988166$$
$$x_{69} = -3385.23867554487$$
$$x_{70} = -1120.11210283123$$
$$x_{71} = 2277.8857177152$$
$$x_{72} = -7349.33295762427$$
$$x_{73} = 2466.64854641277$$
$$x_{74} = -5650.43162631246$$
$$x_{75} = -5084.13202157119$$
$$x_{76} = 9639.7698530287$$
$$x_{77} = -6783.03219947636$$
$$x_{78} = -1875.13203708015$$
$$x_{79} = 7185.79798644413$$
$$x_{80} = -4329.06699021609$$
$$x_{81} = -6027.96498032786$$
$$x_{82} = 4543.06493967891$$
$$x_{83} = 9262.23555158605$$
$$x_{84} = -4895.36562665493$$
$$x_{85} = 3032.94047303143$$
$$x_{86} = 3599.23566805163$$
$$x_{87} = -4706.5993157869$$
$$x_{88} = -2630.18062709538$$
$$x_{89} = -5272.89849150196$$
$$x_{90} = -6216.73172704906$$
Signos de extremos en los puntos:
-pi
(6619.497299625741, 3 - 6621.49729962574 )
-pi
(6241.963673478849, 3 - 6243.96367347885 )
-pi
(-3007.7089148323416, 3 - 3005.70891483234 )
-pi
(7563.331913899711, 3 - 7565.33191389971 )
-pi
(1522.8458074404812, 3 - 1524.84580744048 )
-pi
(-8670.702189815574, 3 - 8668.70218981557 )
-pi
(1711.6033215851749, 3 - 1713.60332158517 )
-pi
(-8104.400976034152, 3 - 8102.40097603415 )
-pi
(7374.564937941762, 3 - 7376.56493794176 )
-pi
(-8859.469292670627, 3 - 8857.46929267063 )
-pi
(5864.430209116776, 3 - 5866.43020911678 )
-pi
(2844.176038071337, 3 - 2846.17603807134 )
-pi
(3976.766925063897, 3 - 3978.7669250639 )
-pi
(8695.934194153888, 3 - 8697.93419415389 )
-pi
(-7538.09992936435, 3 - 7536.09992936435 )
-pi
(5109.363909081075, 3 - 5111.36390908107 )
-pi
(-7915.633940137475, 3 - 7913.63394013748 )
-pi
(2089.123720968191, 3 - 2091.12372096819 )
-pi
(4165.532799835088, 3 - 4167.53279983509 )
-pi
(-5839.19827857882, 3 - 5837.19827857882 )
-pi
(8507.167103633225, 3 - 8509.16710363322 )
-pi
(1900.3628035264567, 3 - 1902.36280352646 )
-pi
(-3762.7694653832486, 3 - 3760.76946538325 )
-pi
(-5461.665028663373, 3 - 5459.66502866337 )
-pi
(7940.865932222735, 3 - 7942.86593222273 )
-pi
(-3574.0039622298036, 3 - 3572.0039622298 )
-pi
(6430.7304681089045, 3 - 6432.7304681089 )
-pi
(6997.031061385308, 3 - 6999.03106138531 )
-pi
(-2818.9445497998954, 3 - 2816.9445497999 )
-pi
(1334.0910936584605, 3 - 1336.09109365846 )
-pi
(9073.468419023122, 3 - 9075.46841902312 )
-pi
(-2441.4172494905906, 3 - 2439.41724949059 )
-pi
(1145.340559543199, 3 - 1147.3405595432 )
-pi
(-9048.23640967437, 3 - 9046.23640967437 )
-pi
(2655.4120299254714, 3 - 2657.41202992547 )
-pi
(5675.6635476865495, 3 - 5677.66354768655 )
-pi
(-7726.8669240837, 3 - 7724.8669240837 )
-pi
(-1497.6157718260438, 3 - 1495.61577182604 )
-pi
(4920.5975001909665, 3 - 4922.59750019097 )
-pi
(8884.7012995941, 3 - 8886.7012995941 )
-pi
(-2063.8927290277884, 3 - 2061.89272902779 )
-pi
(-6594.265339949788, 3 - 6592.26533994979 )
-pi
(-6971.799090552585, 3 - 6969.79909055258 )
-pi
(4731.831173638909, 3 - 4733.83117363891 )
-pi
(3788.00120629721, 3 - 3790.00120629721 )
-pi
(-8481.935102054324, 3 - 8479.93510205432 )
-pi
(-4140.301002546086, 3 - 4138.30100254609 )
-pi
(-6405.498514763926, 3 - 6403.49851476393 )
-pi
(-1308.8616794869647, 3 - 1306.86167948696 )
-pi
(-3951.5351539505546, 3 - 3949.53515395055 )
-pi
(-3196.473643728137, 3 - 3194.47364372814 )
-pi
(5486.896939909349, 3 - 5488.89693990935 )
-pi
(4354.298810341352, 3 - 4356.29881034135 )
-pi
(8318.40002904729, 3 - 8320.40002904729 )
-pi
(5298.1303915154085, 3 - 5300.13039151541 )
-pi
(-4517.833099512, 3 - 4515.833099512 )
-pi
(9451.00269649626, 3 - 9453.00269649626 )
-pi
(-8293.168030418074, 3 - 8291.16803041807 )
-pi
(7752.098912532091, 3 - 7754.09891253209 )
-pi
(6808.264164963073, 3 - 6810.26416496307 )
-pi
(6053.196919184077, 3 - 6055.19691918408 )
-pi
(8129.632971505509, 3 - 8131.63297150551 )
-pi
(3410.470340264852, 3 - 3412.47034026485 )
-pi
(-9237.003539958987, 3 - 9235.00353995899 )
-pi
(-7160.56601068187, 3 - 7158.56601068187 )
-pi
(-2252.654554267021, 3 - 2250.65455426702 )
-pi
(-1686.372859904739, 3 - 1684.37285990474 )
-pi
(3221.7052598816613, 3 - 3223.70525988166 )
-pi
(-3385.2386755448697, 3 - 3383.23867554487 )
-pi
(-1120.1121028312323, 3 - 1118.11210283123 )
-pi
(2277.885717715203, 3 - 2279.8857177152 )
-pi
(-7349.332957624275, 3 - 7347.33295762427 )
-pi
(2466.64854641277, 3 - 2468.64854641277 )
-pi
(-5650.431626312464, 3 - 5648.43162631246 )
-pi
(-5084.132021571186, 3 - 5082.13202157119 )
-pi
(9639.769853028703, 3 - 9641.7698530287 )
-pi
(-6783.0321994763635, 3 - 6781.03219947636 )
-pi
(-1875.1320370801454, 3 - 1873.13203708015 )
-pi
(7185.79798644413, 3 - 7187.79798644413 )
-pi
(-4329.066990216092, 3 - 4327.06699021609 )
-pi
(-6027.964980327864, 3 - 6025.96498032786 )
-pi
(4543.064939678908, 3 - 4545.06493967891 )
-pi
(9262.235551586053, 3 - 9264.23555158605 )
-pi
(-4895.365626654927, 3 - 4893.36562665493 )
-pi
(3032.940473031431, 3 - 3034.94047303143 )
-pi
(3599.2356680516336, 3 - 3601.23566805163 )
-pi
(-4706.599315786899, 3 - 4704.5993157869 )
-pi
(-2630.180627095378, 3 - 2628.18062709538 )
-pi
(-5272.89849150196, 3 - 5270.89849150196 )
-pi
(-6216.731727049065, 3 - 6214.73172704906 )
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
Crece en todo el eje numérico