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:

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 $1$.

Let $f(1) = -2$ and $f'(x) \ge 4.2$ for $1 \le x \le 6$. The smallest possible value of $f(6)$ is:

For the Mean Value Theorem $f(b) - f(a) = (b - a) f'(x_1)$ where $a < x_1 < b$,if $f(x) = 1/x$,then $x_1 = ?$

In which of the following functions is Rolle's theorem applicable?

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) = $