Find the minimum value of $2^{\sin x} + 2^{\cos x}$.
I tried using AM-GM inequality,
$$\frac{2^{\sin x} + 2^{\cos x}}2 ≥ \sqrt{2^{\sin x} 2^{\cos x}}\\ 2^{\sin x} + 2^{\cos x} ≥ 2^{\frac{\sin x + \cos x}2+1}\\$$
This equality holds for $x=\fracπ4$, so I get $2^{\sin x} + 2^{\cos x} = 2^{1 + \frac 1{\sqrt2}}$.
But the minimum is $2^{1-\frac1{\sqrt2}}$.
Edit: I should have considered $x=\frac{5π}4$