The value of $c$ satisfied by Rolle's theorem for the function $f(x) = x^2(1 - x)^2$ on the interval $x \in [0, 1]$ is...

Let $f$ be a function that is continuous and differentiable for all real $x$. If $f(2) = -4$ and $f'(x) \geq 6$ for all $x \in [2, 4]$,then which of the following is true?

Let $f: R \rightarrow R$ be a thrice differentiable function such that $f(0)=0, f(1)=1, f(2)=-1, f(3)=2$ and $f(4)=-2$. Then,the minimum number of zeros of $(3 f^{\prime} f^{\prime \prime} + f f^{\prime \prime \prime})(x)$ is....................

The value of $\left[ \frac{\log (x/e)}{x - e} \right]$ for all $x > e$ is equal to (where $[.]$ denotes the greatest integer function).

Let $f: R \rightarrow R$ be a twice continuously differentiable function such that $f(0) = f(1) = f^{\prime}(0) = 0$. Then:

If $f:[-5,5] \rightarrow R$ is a differentiable function and if $f^{\prime}(x)$ does not vanish anywhere,then prove that $f(-5) \neq f(5).$