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$$\frac{2^{\frac{x - 3}{2}} \log{\left(2 \right)} \operatorname{sign}{\left(2^{\frac{x}{2} - \frac{3}{2}} - 1 \right)}}{2} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = -146.50324745652$$
$$x_{2} = -74.5032474565204$$
$$x_{3} = -100.50324745652$$
$$x_{4} = -106.50324745652$$
$$x_{5} = -128.50324745652$$
$$x_{6} = -112.50324745652$$
$$x_{7} = -98.5032474565204$$
$$x_{8} = -148.50324745652$$
$$x_{9} = -120.50324745652$$
$$x_{10} = -114.50324745652$$
$$x_{11} = -80.5032474565204$$
$$x_{12} = -132.50324745652$$
$$x_{13} = -150.50324745652$$
$$x_{14} = -108.50324745652$$
$$x_{15} = -130.50324745652$$
$$x_{16} = -84.5032474565204$$
$$x_{17} = -92.5032474565204$$
$$x_{18} = -126.50324745652$$
$$x_{19} = -138.50324745652$$
$$x_{20} = -102.50324745652$$
$$x_{21} = -90.5032474565204$$
$$x_{22} = -116.50324745652$$
$$x_{23} = -134.50324745652$$
$$x_{24} = -122.50324745652$$
$$x_{25} = -86.5032474565204$$
$$x_{26} = -110.50324745652$$
$$x_{27} = -124.50324745652$$
$$x_{28} = -156.50324745652$$
$$x_{29} = -118.50324745652$$
$$x_{30} = -96.5032474565204$$
$$x_{31} = -158.50324745652$$
$$x_{32} = -76.5032474565204$$
$$x_{33} = -104.50324745652$$
$$x_{34} = -152.50324745652$$
$$x_{35} = -78.5032474565204$$
$$x_{36} = -136.50324745652$$
$$x_{37} = -94.5032474565204$$
$$x_{38} = -82.5032474565204$$
$$x_{39} = -140.50324745652$$
$$x_{40} = -144.50324745652$$
$$x_{41} = -154.50324745652$$
$$x_{42} = -142.50324745652$$
$$x_{43} = -88.5032474565204$$
$$x_{44} = -160.50324745652$$
Signos de extremos en los puntos:
(-146.50324745652043, 1)
(-74.50324745652043, 0.999999999997839)
(-100.50324745652043, 1)
(-106.50324745652043, 1)
(-128.50324745652043, 1)
(-112.50324745652043, 1)
(-98.50324745652043, 0.999999999999999)
(-148.50324745652043, 1)
(-120.50324745652043, 1)
(-114.50324745652043, 1)
(-80.50324745652043, 0.99999999999973)
(-132.50324745652043, 1)
(-150.50324745652043, 1)
(-108.50324745652043, 1)
(-130.50324745652043, 1)
(-84.50324745652043, 0.999999999999932)
(-92.50324745652043, 0.999999999999996)
(-126.50324745652043, 1)
(-138.50324745652043, 1)
(-102.50324745652043, 1)
(-90.50324745652043, 0.999999999999992)
(-116.50324745652043, 1)
(-134.50324745652043, 1)
(-122.50324745652043, 1)
(-86.50324745652043, 0.999999999999966)
(-110.50324745652043, 1)
(-124.50324745652043, 1)
(-156.50324745652043, 1)
(-118.50324745652043, 1)
(-96.50324745652043, 0.999999999999999)
(-158.50324745652043, 1)
(-76.50324745652043, 0.99999999999892)
(-104.50324745652043, 1)
(-152.50324745652043, 1)
(-78.50324745652043, 0.99999999999946)
(-136.50324745652043, 1)
(-94.50324745652043, 0.999999999999998)
(-82.50324745652043, 0.999999999999865)
(-140.50324745652043, 1)
(-144.50324745652043, 1)
(-154.50324745652043, 1)
(-142.50324745652043, 1)
(-88.50324745652043, 0.999999999999983)
(-160.50324745652043, 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