3A. Solve the equation

$$
\sqrt{1+3 \sin ^{3} x}=3-\sqrt{1-\cos ^{4} x}
$$
Solution. We will write the equation in the form $\sqrt{1+3 \sin ^{3} x}+\sqrt{1-\cos ^{4} x}=3$. We can make the following estimates: $\sin x \leq 1$, hence $\sin ^{3} x \leq 1$, from which $1+3 \sin ^{3} x \leq 4$ and $\cos x \geq 0$, hence $-\cos ^{4} x \leq 0$, from which $1-\cos ^{4} x \leq 1$. Then for the left side of the equation, we have $\sqrt{1+3 \sin ^{3} x}+\sqrt{1-\cos ^{4} x} \leq \sqrt{4}+\sqrt{1}=3$. Equality holds if and only if simultaneously $1+3 \sin ^{3} x=4$ and $1-\cos ^{4} x=1$, i.e., $\sin x=1$ and $\cos x=0$ which is true for $x=\frac{\pi}{2}+2 k \pi$, for $k \in \mathbb{Z}$.