List of top Mathematics Questions on Mean Value Theorem asked in KEAM

Let $f(x)=x^{2}-10x+16$, $x\in\mathbb{R}.$ If $f^{\prime}(c)$ is equal to slope of the straight line joining the points (2,0) and (8,0), then the value of $c$ is

If 2 vectors $ 4\hat{i} + \ell \hat{j} - 6 \hat{k} $ and $ -6 \hat{i} + 12 \hat{j} + 9 \hat{k} $ are collinear, find $\ell$.

If $\sec \theta + \tan \theta = 2 + \sqrt{3}$, then $\sec \theta - \tan \theta$ is:

Let $f(x)=\sqrt{x}$, $4\le x\le16$. If the point $c\in(4,16)$ is such that the tangent line to the graph of $f$ at $x=c$ is parallel to the chord joining $(16,4)$ and $(4,2)$, then the value of $c$ is

A value of $ c $ for which the conclusion of mean value theorem holds for the function $ f(x)=\log_e x $ on the interval $[1,3]$ is

Let $ f(x) $ and $ g(x) $ be two differentiable functions for $ 0 \leq x \leq 1 $ such that $ f(0)=2, g(0)=0, f(1)=6 $. If there exists a real number $ c \in (0,1) $ such that $ f'(c)=2g'(c) $, then $ g(1) $ is equal to

If $y = f(x)$ is continuous on $[0, 6]$, differentiable on $(0, 6)$, $f(0) = -2$ and $f(6) = 16$, then at some point between $x = 0$ and $x = 6$, $f'(x)$ must be equal to:

The function \[ f(x)= \begin{cases} 2x^2-1, & \text{if } 1 \leq x \leq 4 \\ 151-30x, & \text{if } 4 < x \leq 5 \end{cases} \] is not suitable to apply Rolle's theorem since:

The function \[ f(x)= \begin{cases} 2x^2-1, & \text{if } 1 \leq x \leq 4 \\ 151-30x, & \text{if } 4 < x \leq 5 \end{cases} \] is not suitable to apply Rolle's theorem since: