Question

The range of the expression 2^x+2^-x+3^x+3^-x for x∈R is

Answer 1

$\text{Use A.M-G.M inequality,}\\\begin{cases}\frac{2^x+2^{-x}}{2}\ge\sqrt{2^x\cdot2^{-x}}\\\frac{3^x+3^{-x}}{2}\ge\sqrt{3^x\cdot3^{-x}}\end{cases}\\\Rightarrow \begin{cases}2^x+2^{-x}\ge\sqr2\\3^x+3^{-x}\ge2\end{cases}\\\text{The minimum exists at }x=0\\\text{ in both cases so add both inequalities:}\\2^x+2^{-x}+3^x+3^{-x}\ge4\\\text{Therefore,}\\Range=[4,\infty)$