Question

If Rolle's theorem is applied to $f\left(x\right)=\sin x+\cos x+1;x\in [\pi ,\frac{3\pi }{2}]$, then the value of $c$ is ______.

Answer 1

Given,

$f\left(x\right)=\sin x+\cos x+1;x\in [\pi ,\frac{3\pi }{2}]$

We have, $f\left(\pi \right)=0-1+1=0$.

Again $f\left(\frac{3\pi }{2}\right)=-1+0+1=0$.

Also $f\left(x\right)$ is continuous in the given interval and is differentiable in $(\pi ,\frac{3\pi }{2})$.

So all the conditions of Rolle's theorem are satisfied by $f\left(x\right)$ in the given interval.

Then $\exists$ a real number $c$ $\in (\pi ,\frac{3\pi }{2})$ such that,

$f^{\prime }\left(c\right)=0$

or, $\cos c-\sin c=0$

or, $\tan c=1$

or, $c=n\pi +\frac{\pi }{4}$.

Since $c\in$$(\pi ,\frac{3\pi }{2})$. so $c=\frac{3\pi }{4}$.