[ [trigonometric inequalities]

For what values of $x$ does the inequality $2^{\sin ^{2} x}+2^{\cos ^{2} x} \geq 2 \sqrt{2}$ hold?
Both terms on the left side are positive. By the Cauchy-Schwarz inequality

$$
2^{\sin ^{2} x}+2^{\cos ^{2} x} \geq 2 \sqrt{2^{\sin ^{2} x} \cdot 2^{\cos ^{2} x}}=2 \sqrt{2^{\sin ^{2} x+\cos ^{2} x}}=2 \sqrt{2}
$$

## Answer

For all real values of $x$.