5. If $0<x<\frac{\pi}{2}$, and $\frac{\sin ^{4} x}{9}+\frac{\cos ^{4} x}{4}=\frac{1}{13}$, then the value of $\tan x$ is ( ).
(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) 1
(D) $\frac{3}{2}$
5. D.

By the Cauchy-Schwarz inequality, we have
$$
(9+4)\left(\frac{\sin ^{4} x}{9}+\frac{\cos ^{4} x}{4}\right) \geqslant\left(\sin ^{2} x+\cos ^{2} x\right)^{2} \text {. }
$$

From the equality condition, we get
$$
\frac{\sin ^{4} x}{81}=\frac{\cos ^{4} x}{16} \Rightarrow \tan x=\frac{3}{2} .
$$