Question

Let $f\left(x\right)=2+\cos x$ for all real x. STATEMENT-1: For each real t, there exists a point c in $[t,t+\pi ]$ such that $f^{\prime }\left(c\right)=0$ because STATEMENT-$2:f\left(t\right)=f(t+2\pi )$ for each real t.

• A. statement -1 is True, Statement -2 is True; Statement-2 is a correct explanation for statement-1
• B. Statement -1 is True, Statement-2 is True; Statement-2 is Not a correct explanation for statement -1
• C. Statement - 1 is True, Statement - 2 is False
• D. Statement -1 is False, Statement -2 is True

Answer 1

The correct option is B Statement -1 is True, Statement-2 is True; Statement-2 is Not a correct explanation for statement -1

$f^{\prime }\left(c\right)=\sin \left(c\right)=0$
Or
$c=n\pi$
Or
$t=\frac{(2n-1)\pi }{2}$.
Now period of $\sin \left(x\right)$ and $\cos \left(x\right)$ is $2\pi$.
Hence $0<t<\pi$.
Thus we get atleast one integral multiple of $\pi$ in between $[t,t+\pi ]$.
Hence both assertion and reasons are correct and reason is the correct explanation.