Explanation

(a)
Let y$= \sin^{3}{x + \cos^{3}x}$
$\frac{dy}{dx} = 3\sin^{2}{x\cos{x - 3\cos^{2}x\sin x}}$
$= 3\sin{x\cos{x\ \left( \sin{x - \cos x} \right)}}$
Put, $\frac{dy}{dx} = 0,\ \ \ \ \ 3\sin{x\cos x\left( \sin{x - \cos x} \right) = 0}$
$\Rightarrow \ \ \ \ \ \ \ x = 0\ \frac{\pi}{2}\$or $\frac{\pi}{4}$
Now, $y$ has its maximum value at $x = 0$ or $\frac{\pi}{2}$ and $y_{\max\ } = 1$