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Let $f(x)=2+\cos x$ for all real $x$.
$STATEMENT -1$ : For each real $\mathrm{t}$, there exists a point $\mathrm{c}$ in $[\mathrm{t}, \mathrm{t}+\pi]$ such that $\mathrm{f}^{\prime}(\mathrm{c})=0$. because
$STATEMENT -2$: $f(t)=f(t+2 \pi)$ for each real $t$.
Statement -$1$ is True, Statement -$2$ is True; Statement-$2$ is a correct explanation for Statement-$1$
Statement -$1$ is True, Statement - $2$ is True; Statement-$2$ is $NOT$ a correct explanation for Statement-$1$
Statement -$1$ is True, Statement -$2$ is False
Statement -$1$ is False, Statement -$2$ is True
Solution
$f(x)=2+\cos x \forall x \in R$
Statement : $1$
There exists a point $c \in[t, t+\pi]$ where $f^{\prime}(c)=0$
Hence, statement 1 is true.
Statement $2$ :
$\mathrm{f}(\mathrm{t})=\mathrm{f}(\mathrm{t}+2 \pi)$ is true.
But statement $2$ is not a correct explanation for statement $1$.
