List of top Mathematics Questions on Mean Value Theorem

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.

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

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: \[ \begin{array}{|c|c|} \hline \text{Class Interval} & \text{Frequency} \\ \hline 0-6 & 1 \\ 6-12 & 2 \\ 12-18 & 3 \\ 18-24 & 2 \\ 24-30 & 1 \\ \hline \end{array} \]
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 :
The value of c∈(0,2) satisfying the mean value theorem for the function f(x)=x(x-1)² on [0,2] is:
The value of $c$ in Lagrange's theorem for the function $f(x) = \log (\sin \, x)$ in the interval $\left[ \frac{\pi}{6} , \frac{5 \pi}{6} \right]$ is
The value of $c$ from the Lagrange�s mean value theorem for which $f(x) = \sqrt{25 - x^2}$ in $[1,5]$, is
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:
The value of C in (0, 2) satisfying the mean value theorem for the function \(f(x)=x(x-1)^2, x∈[0,2]\) is equal to