Let $f$ be differentiable for all $x$. If $f(1) = -2$ and $f'(x) \ge 2$ for $x \in [1, 6]$,then

Let $f$ be continuous on $[1, 5]$ and differentiable in $(1, 5).$ If $f(1)=-3$ and $f'(x) \ge 9$ for all $x \in (1, 5)$,then which of the following is true?

Let $g: R \rightarrow R$ be a non-constant twice differentiable function such that $g^{\prime}\left(\frac{1}{2}\right)=g^{\prime}\left(\frac{3}{2}\right)$. If a real-valued function $f$ is defined as $f(x)=\frac{1}{2}[g(x)+g(2-x)]$,then:

If $f$ and $g$ are differentiable functions in $[0, 1]$ satisfying $f(0) = 2$,$g(1) = 2$,$g(0) = 0$,and $f(1) = 6$,then for some $c \in (0, 1)$:

If $f(x)$ satisfies the conditions of Rolle's theorem in $[1, 2]$ and $f(x)$ is continuous in $[1, 2]$,then $\int_1^2 f'(x) dx$ is equal to

If $f(x) = x^\alpha \log x$ and $f(0) = 0$,then the value of $\alpha$ for which Rolle's theorem can be applied in $[0, 1]$ is