Question

Find the greatest and the least values of the following functions.
$f\left(x\right)=(2^x+2^{-x})/In2$ on the interval [-1, 2].

Answer 1

$f\left(x\right)=\frac{2^x+2^{-x}}{\ln 2}$

$f^{\prime }\left(x\right)=2^x\ln 2-2^{-x}\ln 2=0\Rightarrow x={\log }_2\sqrt{2}$

$f^{\prime \prime }\left(x\right)=\left(\ln 2{)}^2\right(2^x+2^{-x})>0$

$f\left({\log }_2\sqrt{2}\right)=(\sqrt{2}+\frac{1}{\sqrt{2}})\frac{1}{\ln 2}\left(Minimumvalue\right)$

The function increases continously before and after that,it has no maximum value.