List of top Mathematics Questions on Mean Value Theorem

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
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
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
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 :
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
The value of c∈(0,2) satisfying the mean value theorem for the function f(x)=x(x-1)² on [0,2] is:
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 value of \(c\) prescribed by Lagrange's mean value theorem, when \(f(x) = \sqrt{x^2 - 4}\), \(a = 2\) and \(b = 3\), is
What are the values of \( c \) for which Rolle’s theorem for the function \( f(x) = x^3 - 3x^2 + 2x \) in the interval \( [0,2] \) is verified?

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:

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:

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: