If the function $f(x) = x(x + 3) e^{-x/2}$ satisfies Rolle's theorem in the interval $[-3, 0]$,then find the value of $c$.

For what value of $c$ does the conclusion of the Mean Value Theorem hold for the function $f(x) = \log_e x$ on the interval $[1, 3]$?

Let $f(x) = \begin{cases} \frac{x^p}{(\sin x)^q} & \text{if } 0 < x \leq \frac{\pi}{2} \\ 0 & \text{if } x = 0 \end{cases}$ where $p, q \in \mathbb{R}$. Then, Lagrange's Mean Value Theorem is applicable to $f(x)$ in the closed interval $[0, \frac{\pi}{2}]$ if:

Consider the function $f(x) = \begin{cases} x \sin \frac{\pi}{x} & \text{for } x > 0 \\ 0 & \text{for } x = 0 \end{cases}$. Then the number of points in $(0, 1)$ where the derivative $f'(x)$ vanishes is:

The equation $2^x+5^x=3^x+4^x$ has

Consider the 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-$2$: Rolle's theorem can be applied to the function $g(x)$ in the interval $[0, 1]$.