Consider the function $f(x) = 8x^2 - 7x + 5$ on the interval $[-6, 6]$. The value of $c$ that satisfies the conclusion of the Mean Value Theorem is:

If $c$ is a point at which Rolle's theorem holds for the function $f(x) = \log_{e}\left(\frac{x^{2}+\alpha}{7x}\right)$ in the interval $[3, 4]$,where $\alpha \in R$,then $f''(c)$ is equal to

If $L.M.V.T.$ is true for $f(x) = x(x-1)(x-2)$ on the interval $x \in [0, 1/2]$,then find the value of $C$.

The function $f(x) = x^3 - 6x^2 + ax + b$ satisfies the conditions of Rolle's theorem in $[1, 3]$. Then the values of $a$ and $b$ are respectively

Let $f:(a, b) \rightarrow R$ be a twice differentiable function such that $f(x) = \int_{a}^{x} g(t) \, dt$ for a differentiable function $g(x)$. If $f(x) = 0$ has exactly five distinct roots in $(a, b)$,then $g(x) g'(x) = 0$ has at least:

Let $f:[1,3] \rightarrow R$ be continuous and differentiable in $(1,3)$ such that $f^{\prime}(x)=[f(x)]^2+4$ for all $x \in (1,3)$. Then: