Mathematics - Application of Derivative Question with Solution | TestHub

Because $a=\frac{4}{\sin x}+\frac{1}{1-\sin x}$, where $a$ is least.

$$\frac{da}{dx}=\left(\frac{-4}{{\sin }^{2}x}+\frac{1}{(1-\sin x{)}^2}\right)\cos x=0$$

Because $\cos x\ne 0$, we have $\sin x=2/3$.

For $\sin x=2/3$, $\frac{d^{2}a}{d{x}^2}=45>0$.

Thus, $a=\frac{4}{2/3}+\frac{1}{1-2/3}=6+3=9$.