The value of $c$ for the Lagrange's mean value theorem for $f(x)=\sqrt{x^2-x}, x \in[1,4]$ is

Consider the function $f(x) = |x - 2| + |x - 5|, x \in R$.
Statement-$1$: $f'(4) = 0$.
Statement-$2$: $f$ is continuous in $[2, 5]$,differentiable in $(2, 5)$ and $f(2) = f(5)$.

Let $f:[a, b] \rightarrow R$ be differentiable on $[a, b]$ and $k \in R$. Let $f(a)=0=f(b)$. Also let $J(x)=f'(x)+k f(x)$. Then

If $f(x)$ is a twice differentiable polynomial function such that $f(1) = 1, f(2) = 4, f(3) = 9$,then:

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

Let $f(x)$ and $g(x)$ be two functions which are defined and differentiable for all $x \ge x_0$. If $f(x_0) = g(x_0)$ and $f'(x) > g'(x)$ for all $x > x_0$,then: