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$$2^{x} \log{\left(2 \right)} \operatorname{sign}{\left(2^{x} - 4 \right)} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -74.176760093132$$
$$x_{2} = -60.176760093132$$
$$x_{3} = -64.176760093132$$
$$x_{4} = -112.176760093132$$
$$x_{5} = -54.176760093132$$
$$x_{6} = -50.176760093132$$
$$x_{7} = -52.176760093132$$
$$x_{8} = -110.176760093132$$
$$x_{9} = -88.176760093132$$
$$x_{10} = -82.176760093132$$
$$x_{11} = -66.176760093132$$
$$x_{12} = -46.176760093132$$
$$x_{13} = -114.176760093132$$
$$x_{14} = -80.176760093132$$
$$x_{15} = -42.176760093132$$
$$x_{16} = -106.176760093132$$
$$x_{17} = -104.176760093132$$
$$x_{18} = -62.176760093132$$
$$x_{19} = -108.176760093132$$
$$x_{20} = -94.176760093132$$
$$x_{21} = -72.176760093132$$
$$x_{22} = -124.176760093132$$
$$x_{23} = -56.176760093132$$
$$x_{24} = -68.176760093132$$
$$x_{25} = -98.176760093132$$
$$x_{26} = -44.176760093132$$
$$x_{27} = -120.176760093132$$
$$x_{28} = -100.176760093132$$
$$x_{29} = -128.176760093132$$
$$x_{30} = -48.176760093132$$
$$x_{31} = -92.176760093132$$
$$x_{32} = -40.176760093132$$
$$x_{33} = -118.176760093132$$
$$x_{34} = -58.176760093132$$
$$x_{35} = -96.176760093132$$
$$x_{36} = -78.176760093132$$
$$x_{37} = 2$$
$$x_{38} = -70.176760093132$$
$$x_{39} = -116.176760093132$$
$$x_{40} = -76.176760093132$$
$$x_{41} = -126.176760093132$$
$$x_{42} = -90.176760093132$$
$$x_{43} = -84.176760093132$$
$$x_{44} = -130.176760093132$$
$$x_{45} = -102.176760093132$$
$$x_{46} = -86.176760093132$$
$$x_{47} = -122.176760093132$$
Signos de extremos en los puntos:
(-74.17676009313203, 4)
(-60.176760093132025, 4)
(-64.17676009313203, 4)
(-112.17676009313203, 4)
(-54.176760093132025, 4)
(-50.176760093132025, 4)
(-52.176760093132025, 4)
(-110.17676009313203, 4)
(-88.17676009313203, 4)
(-82.17676009313203, 4)
(-66.17676009313203, 4)
(-46.176760093132025, 3.99999999999999)
(-114.17676009313203, 4)
(-80.17676009313203, 4)
(-42.176760093132025, 3.9999999999998)
(-106.17676009313203, 4)
(-104.17676009313203, 4)
(-62.176760093132025, 4)
(-108.17676009313203, 4)
(-94.17676009313203, 4)
(-72.17676009313203, 4)
(-124.17676009313203, 4)
(-56.176760093132025, 4)
(-68.17676009313203, 4)
(-98.17676009313203, 4)
(-44.176760093132025, 3.99999999999995)
(-120.17676009313203, 4)
(-100.17676009313203, 4)
(-128.17676009313203, 4)
(-48.176760093132025, 4)
(-92.17676009313203, 4)
(-40.176760093132025, 3.9999999999992)
(-118.17676009313203, 4)
(-58.176760093132025, 4)
(-96.17676009313203, 4)
(-78.17676009313203, 4)
(2, 0)
(-70.17676009313203, 4)
(-116.17676009313203, 4)
(-76.17676009313203, 4)
(-126.17676009313203, 4)
(-90.17676009313203, 4)
(-84.17676009313203, 4)
(-130.17676009313203, 4)
(-102.17676009313203, 4)
(-86.17676009313203, 4)
(-122.17676009313203, 4)
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:
$$x_{1} = 2$$
La función no tiene puntos máximos
Decrece en los intervalos
$$\left[2, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, 2\right]$$