If $f(x)$ is differentiable on the interval $[2, 5]$ such that $f(2) = 1/5$ and $f(5) = 1/2$,then there exists a number $c$ such that $2 < c < 5$ and $f'(c) = \dots$

Let $f:[1,3] \rightarrow R$ be a continuous function that is differentiable in $(1,3)$ and $f^{\prime}(x)=|f(x)|^{2}+4$ for all $x \in(1,3).$ Then,

If for $f(x) = 2x - x^2$,Lagrange's Mean Value Theorem satisfies in $[0, 1]$,then the value of $c \in [0, 1]$ is

If $f:[-5,5] \rightarrow R$ is a differentiable function and if $f^{\prime}(x)$ does not vanish anywhere,then prove that $f(-5) \neq f(5).$

The position of a moving car at time $t$ is given by $f(t) = at^{2} + bt + c, t > 0,$ where $a, b,$ and $c$ are real numbers greater than $1.$ Then the average speed of the car over the time interval $[t_{1}, t_{2}]$ is attained at the point

If a function $f$ is differentiable on $R$ such that $f^{\prime}(x) \leq 4$ for all $x \in R$; and if $f(2)=-6$ and $f(6)=8$,then the value of $f(4)$ belongs to the interval