Verify Rolle's theorem for the function $y=x^{2}+2$ on the interval $[-2, 2]$.

If $f(x)$ is a twice differentiable polynomial function such that $f(1) = 1, f(2) = 4, f(3) = 9$,then:

Let $f(x) = \begin{cases} \frac{x^p}{(\sin x)^q} & \text{if } 0 < x \leq \frac{\pi}{2} \\ 0 & \text{if } x = 0 \end{cases}$ where $p, q \in \mathbb{R}$. Then,Lagrange's Mean Value Theorem is applicable to $f(x)$ in the closed interval $[0, \frac{\pi}{2}]$ if:

Let $f$ be a twice differentiable function on $(1,6)$. If $f(2)=8$,$f'(2)=5$,$f'(x) \geq 1$ and $f''(x) \geq 4$ for all $x \in (1,6)$,then:

Let $f:[a, b] \rightarrow R$ be continuous in $[a, b]$,differentiable in $(a, b)$ and $f(a)=0=f(b)$. Then

For what value of $c$ does the conclusion of the Mean Value Theorem hold for the function $f(x) = \log_e x$ on the interval $[1, 3]$?