AM-GM Basic 2

Algebra Level 2

If the minimum value of $f(x)=2^{a+x}+2^{a-x}$ is equal to $8$ , what is the value of $a$ ?

1 solution

Junghwan Han
Aug 18, 2019

Since $\displaystyle{2^{a+x}>0}$ and $\displaystyle{2^{a-x}>0}$ for all real $x$

Due to the Arithmetic Mean-Geometric Mean Inequality, $2^{a+x}+2^{a-x}\geq2\sqrt{2^{a+x}2^{a-x}}=2\sqrt{2^{2a}}=2^{a+1}$ Equality holds when $x=0$

Therefore the minimum value of $f(x)$ is $\displaystyle{2^{a+1}}$ $2^{a+1}=8$ $2^{a+1}=2^3$ $\therefore a=2$

@Junghwan Han , \displaystyle is not necessary here.

Chew-Seong Cheong - 1 year, 9 months ago