List of top Mathematics Questions on Mean Value Theorem

If \(f(x)\) satisfy the condition of Rolle's Theorm in \([1,2]\) and \(f(x)\) is continuous in \([1,2]\) then \(\int _{1}^{2}f'(x)dx\) is equal to _____
If Rolle's theorem holds for the function \(f(x)=x^{2}-px+10\), \(x\in [0,4]\) with \(c=2\), then \(p=\)
In the Lagrange's Mean Value Theorem, \(f(b)-f(a)=(b-a)f'(c);\ (a\lt c\lt b)\). If \(a=2,b=4\) and \(f(x)=\sqrt{x^{2}-4}\), then the value of \(c\) is
The value of \(c\) for which the function \(f(x)=\log \left(\frac{x^{2}+6}{5x}\right)\) satisfies Rolle's theorem in the interval \([2,3]\) is ____
If Rolle's theorem holds for the function \(f(x)=x^{3}+px^{2}+qx+5,x\in [1,3]\) with \(c=2+\frac{1}{\sqrt{3}}\), then the values of \(p\) and \(q\) are respectively...
For which of the following functions is Rolle's theorem not applicable?
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: \[ 9c^2-12c = \ ? \] 

The function \(f(x) = x(x+3)e^{-(\frac{1}{2})x}\) satisfies all the conditions of Rolle's theorem in \([-3,0]\), then \(c =\)
Rolle's theorem holds for monic quadratic polynomial \(f(x)\) on the interval \([α,α+3]\) where \(f(α) = 0\). Similarly, \(g(x) = f(x)+2\) also follows Rolle's theorem in the interval \([β,3]\) where \(g(3) = 0\), such that the value of \(c\) is the same for both \(f(x)\) and \(g(x)\). Then the value of \((f\circ g)(α)\) is...
The value of \(c\) satisfied by the Rolle's theorem for the function \(f(x) = x^2(1-x)^2\), \(x\in [0,1]\) is...
If the function \(f(x) = ax^2+bx+sinx\) satisfies all the conditions of Rolle's theorem on \([0,π]\) and the slope of the tangent to the curve \(y = f(x)\) at \(x = \frac{π}{4}\) is zero, then \(a-b =\)
If the function \(f(x) = ax^3-bx^2-8x-4\) satisfies Roll's theorem in \([1,3]\), if \(f^'(2) = 0\) then \(a-b\) is equal to...
If Rolle's theorem is applicable for the function \(f(x) = log(\frac{x^2+a}{x})\) on \([3,4]\) with \(c\in (3,4)\) such that \(f^'(c) = 0\), then the value of \(f^{''}(c)\) is
Let \(f(x)\) be the differentiable function for all \(x\) such that \(f^'(x)\leq 5\) and \(f(1) = 4\). The maximum value of \(f(5)\) is...
If the function \(f(x) = ax^3+bx^2+11x-6\), defined on \([1,3]\), satisfies all the conditions of Rolle's theorem for \(c = 2+\frac{1}{\sqrt{3}}\), then
Which of the following function satisfies hypotheses and the conclusion of the Lagrange Mean Value Theorem
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 \(y=x^3\) are in the interval \([-2,2]\), where the slope of the tangents can be obtained by mean value theorem for the interval \([-2,2]\) are

If Rolle's theorem holds for the function \(x^3+\mathrm{a} x^2+\mathrm{b} x, 1 \leq x \leq 2\) at the point \(\frac{4}{3}\), then the values of \(a\) and \(b\) are respectively

If the function \(\mathrm{f}(x)=x(x+3) \mathrm{e}^{-\frac{x}{2}}\) satisfies all the conditions of Rolle's theorem in \([-3,0]\), then 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