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: $\quad \frac{1}{2}$
B: $\frac{2}{3}$
C: 1
$\mathrm{D}: \frac{3}{2}$
Answer D.
Analysis By Cauchy-Schwarz inequality, $(9+4)\left(\frac{\sin ^{4} x}{9}+\frac{\cos ^{4} x}{4}\right) \geq\left(\sin ^{2} x+\cos ^{2} x\right)^{2}$, and by the equality condition we have
$$
\frac{\sin ^{4} x}{81}=\frac{\cos ^{4} x}{16} \Rightarrow \tan x=\frac{3}{2}
$$