*(This item, "The minimum value of 2^ x+2^ x", is a trigonometry/AM-GM question, not a Sta…

Question

*(This item, "The minimum value of $2^{\sin x}+2^{\cos x}$", is a trigonometry/AM-GM question, not a Statistics topic item --- it appears misplaced under this heading in the source PDF. It is retained here for completeness of the source document.)*

Accepted Answer

By AM--GM: $2^{\sin x}+2^{\cos x} \ge 2\cdot2^{\frac{\sin x+\cos x}{2}}$. Since $-\sqrt2\le\sin x+\cos x\le\sqrt2$, the minimum of $2^{\frac{\sin x+\cos x}{2}}$ is $2^{-1/\sqrt2}$.

$$2^{\sin x}+2^{\cos x} \ge 2\cdot2^{-1/\sqrt2} = 2^{1-\frac{1}{\sqrt2}}$$

Solutions · 1