List of top Mathematics Questions on Mean Value Theorem

Which of the following function satisfies hypotheses and the conclusion of the Lagrange Mean Value Theorem
If the function \[ f(x)=x^3+bx^2+ax \] satisfies the conditions of Rolle's theorem in \([1,3]\) with \[ c=2+\frac{1}{\sqrt3}, \] then \((a,b)=\)

If \(c \in (1,3)\) satisfies Lagrange’s Mean Value Theorem for \[ f(x)=x^3-2x^2+x-1 \] on \([1,3]\), then find: \[ 9c^2-12c = \ ? \] 

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 \(\ell\).
The abscissae of the points of the curve \[ y = x^3 \quad \text{are in the interval} \quad [-2, 2], \] where the slope of the tangents can be obtained by the mean value theorem for the interval \( [-2, 2] \), are

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

If $a^2 x^4 + b^2 y^4 = c^6$, then the maximum value of $xy$ is:
The number of all the values of \( x \) for which the function \[ f(x) = \sin x + \frac{1 - \tan^2 x}{1 + \tan^2 x} \] attains its maximum value on \( [0, 2\pi] \).
If Rolle’s theorem is applicable for the function \( f(x) \) defined by \( f(x) = x^3 + Px - 12 \) on \( [0,1] \), then the value of \( C \) of the Rolle's theorem is:

If the function f(x) = \(\sqrt{x^2 - 4}\)  satisfies the Lagrange’s Mean Value Theorem on \([2, 4]\), then the value of \( C \) is} 

The value of $C$ in $(0, 2)$ satisfying the mean value theorem for the function $f(x) = x(x - 1)^2$, $x \in [0, 2]$ is equal to:}
The value \( c \) of Lagrange’s Mean Value Theorem for \( f(x) = e^x + 24 \) in \( [0,1] \) is:
In each of the following options, a function and an interval are given. Choose the option containing the function and the interval for which Lagrange’s Mean Value Theorem is not applicable.
The mean of five observations is 4 and their variance is 5.2. If three of these observations are 1, 2, and 6, then the other two observations are:
The value(s) of $ c \in (1, 2) $, where the conclusion of Lagrange’s M.V.T. is satisfied for the function $ f(x) = x^2 + 3x + 2 $ in $[1,2]$, is/are:
Mean deviation about the mean for the following data is: \[ \begin{array}{|c|c|} \hline \text{Class Interval} & \text{Frequency} \\ \hline 0-6 & 1 \\ 6-12 & 2 \\ 12-18 & 3 \\ 18-24 & 2 \\ 24-30 & 1 \\ \hline \end{array} \]
The value of c of Lagrange's mean value theorem for $f(x)=\sqrt{25-x^{2}}$ on $[1,5]$ is
Let A₁ and A₂ be two arithmetic means and G1, G2, G3 be three geometric means of two distinct positive numbers. Then $ G_{1}^{4} + G_{2}^{4}+ G_{3}^{4} +G_{1}^{2}G_{3}^{2} $ is equal to
If \( f(x) \) is a differentiable function, \( f'(x) \geq 5 \) for \( x \in [2,6] \), \( f(2) = 4 \) and \( f(3) = 15 \), then a possible value of \( f(6) \) is:
Let the mean and the variance of 5 observations x1, x2, x3, x4, x5 be \(\frac{24}{5} \) and \(\frac{194}{25}\), respectively. If the mean and variance of the first 4 observations are \(\frac{7}{2}\) and a, respectively, then (4a + x5) is equal to
If Rolle's theorem holds for the function f(x) = x³ - ax² + bx - 4, x ∈ [1, 2] with f'(4/3) = 0, then ordered pair (a, b) is equal to :
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