If $f(x) = \sqrt{x}$ and $g(x) = \frac{1}{\sqrt{x}}$ for $x \in [3, 12]$,then the value of $c \in (3, 12)$ for which $\frac{f^{\prime}(c)}{g^{\prime}(c)} = \frac{f(12) - f(3)}{g(12) - g(3)}$ holds,is

If $f(x)$ is a differentiable function,$f^{\prime}(x) \geq 5$ for all $x \in [2, 6]$,$f(2) = 4$ and $f(3) = 15$,then a possible value of $f(6)$ is:

Verify the Mean Value Theorem for the function $f(x) = x^{2}$ in the interval $[2, 4]$.

Verify Rolle's Theorem for the function $f(x)=x^{2}+2x-8, x \in[-4,2]$

Consider a quadratic equation $ax^2 + bx + c = 0,$ where $2a + 3b + 6c = 0$ and let $g(x) = a\frac{x^3}{3} + b\frac{x^2}{2} + cx.$
Statement $1:$ The quadratic equation has at least one root in the interval $(0, 1).$
Statement $2:$ The Rolle's theorem is applicable to function $g(x)$ on the interval $[0, 1].$

Let $f$ be continuous on $[1, 5]$ and differentiable in $(1, 5).$ If $f(1)=-3$ and $f'(x) \ge 9$ for all $x \in (1, 5)$,then which of the following is true?