Question

The least value of '$a$' for which equation $\frac{4}{\sin x}+\frac{1}{1-\sin x}=a$ has at least one solution in the interval $[0,\frac{\pi }{2}]$ is

• A. $3$
• B. $11$
• C. $7$
• D. $9$

Answer 1

The correct option is D $9$

Given $f\left(x\right)=\frac{4}{\sin x}+\frac{1}{1-\sin x}=a$
by $f^{\prime }\left(x\right)$ we conclude that
f(x) decrease for $\sin x\in (0,2/3)$ and f(x) increase for $\sin x\in (2/3,1)$
hence it will take least value
when $\sin x=2/3\Rightarrow a_{min}=f\left(2/3\right)=9$