Let $f$ be a function which is continuous and differentiable for all $x$. If $f(1) = 1$ and $f^{\prime}(x) \leq 5$ for all $x$ in $[1, 5]$,then the maximum value of $f(5)$ is

Let $a, b, c$ be real numbers such that $2a + 3b + 6c = 0$ and $g(x) = ax^2 + bx + c = 0$ has at least one root in the interval $(1, 2)$. If a function $f: [1, 2] \rightarrow \mathbb{R}$ for which Rolle's Theorem holds is such that $f(x)$ is a primitive of $g(x)$,then $f(x) = $

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:

Rolle's theorem is applicable for the function $f(x) = x^2 - 4$ in which of the following intervals?

If $2a + 3b + 6c = 0$ and $a, b, c \in \mathbb{R}$,then the equation $ax^2 + bx + c = 0$ has at least one root between $0$ and $1$.

The value of $c$ for which the Lagrange's Mean Value Theorem $(LMVT)$ is applicable for the function $f(x) = x(x+3)(x-2)$ in the interval $[-1, 4]$ is: