Let $f:[a, b] \rightarrow R$ be continuous in $[a, b]$, differentiable in $(a, b)$ and $f(a)=0=f(b)$. Then

Let $f$ be any continuously differentiable function on $[a, b]$ and twice differentiable on $(a, b)$ such that $f(a)=f^{\prime}(a)=0$ and $f(b)=0$. Then:

If the function $f(x) = x(x+3)e^{-x/2}$ satisfies all the conditions of Rolle's theorem in $[-3, 0]$, then a root of $f'(x) = 0$ is

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

Consider $f(x) = \int\limits_0^x {\left( {t + \frac{1}{t}} \right)\,dt}$ and $g(x) = f'(x)$ for $x \in \left[ {\frac{1}{2}, 3} \right]$. If $P$ is a point on the curve $y = g(x)$ such that the tangent to this curve at $P$ is parallel to a chord joining the points $\left( {\frac{1}{2}, g\left( {\frac{1}{2}} \right)} \right)$ and $(3, g(3))$ of the curve,then the coordinates of the point $P$ are:

Let $f$ be differentiable for all $x$. If $f(1) = -2$ and $f'(x) \ge 2$ for $x \in [1, 6]$,then