Question

The greatest value of $si{n}^{3}x+co{s}^{3}x$ is

• A. $1$
• B. $2$
• C. $\sqrt{2}$
• D. $\sqrt{3}$

Answer 1

Answer: (A)

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