Question

For the function $f\left(x\right)=e^{\cos x}$, Rolle’s theorem is

• A. applicable when$\frac{\pi }{2}\le x\le \frac{3\pi }{2}$
• B. applicable when$0\le x\le \frac{\pi }{2}$
• C. applicable when $0\le x\le \pi$
• D. applicable when$\frac{\pi }{4}\le x\le \frac{\pi }{2}$

Answer 1

The correct option is A

applicable when$\frac{\pi }{2}\le x\le \frac{3\pi }{2}$

Applicability for Rolle's theorem :

Step1: Defining Rolle's theorem
A function $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$ such that $f\left(a\right)=f\left(b\right)$, then $f\prime \left(x\right)=0$ for some $x$ with$a\le x\le b$

Step2: Verifying Rolle's theorem,

Taking,

$\begin{array}{rcl}\frac{\pi }{2}& \le & x\le \frac{3\pi }{2}\\ f\left(\frac{\pi }{2}\right)& =& e^{\cos \frac{\pi }{2}}=1\end{array}$

Similarly for,

$\begin{array}{rcl}f\left(\frac{3\pi }{2}\right)& =& e^{\cos \frac{3\pi }{2}}=1\\ f\left(\frac{\pi }{2}\right)& =& f\left(\frac{3\pi }{2}\right)\end{array}$

Hence, correct option is $\mathbf{\left(}\mathit{A}\mathbf{\right)}$