### 3.486 Find the maximum value of the expression

$A=\frac{1}{\sin ^{4} \alpha+\cos ^{4} \alpha}$ for $0 \leq \alpha \leq \frac{\pi}{2}$.
Solution. $\sin ^{4} \alpha+\cos ^{4} \alpha=1-2 \sin ^{2} \alpha \cdot \cos ^{2} \alpha=1-\frac{\sin ^{2} 2 \alpha}{2}=\frac{1+\cos ^{2} 2 \alpha}{2} \Rightarrow$
$\Rightarrow A=\frac{2}{1+\cos ^{2} 2 \alpha}$. From this, it is clear that $A$ takes its maximum value when $1+\cos ^{2} 2 \alpha$ takes its minimum value, i.e., when $\cos 2 \alpha=0$ or $\alpha=\frac{\pi}{4}$.

Answer: 2 when $\alpha=\frac{\pi}{4}$.