Question

The number of local extremum points of the function $f\left(x\right)=e^{\sin x}+2\sin x+x-4in\left(0,\frac{\pi }{2}\right)$ is
[Note: $\left[k\right]$ and sgn$\left(k\right)$ denotes greater integer function and signum function of $k$ respectively.]

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (A)

$0$

$f\left(x\right)=e^{\sin x}+2\sin x-4,x\in (0,\Pi /2)f^{\prime }\left(x\right)=e^{\sin x}\left(\cos x\right)+2\cos \left(x\right)+1$

For extremum points, $f^{\prime }\left(x\right)=0$

since $x\in (0,\pi /2)\sin x$ & $\cos x$ are both positive

$e^{\sin x}$ will always be positive

Therefore $f^{\prime }\left(x\right)\ne 0$

Hence, there are no extremum points.