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...

Values of $c$ as per Rolle's theorem for $f(x)=\sin x+\cos x+6$ on $[0, 2\pi]$ are

Let $f :[0,1] \rightarrow R$ be a twice differentiable function in $(0,1)$ such that $f(0)=3$ and $f(1)=5$. If the line $y=2x+3$ intersects the graph of $f$ at only two distinct points in $(0,1)$,then the least number of points $x \in(0,1)$,at which $f^{\prime\prime}(x)=0$,is $......$

For the curve $y = x^3$ in the interval $[-2, 2]$,find the abscissae of the points where the slope of the tangent is equal to the slope of the secant line passing through the endpoints of the interval,as per the Mean Value Theorem.

Examine if Rolle's Theorem is applicable to the function $f(x) = x^{2} - 1$ for $x \in [1, 2]$. Can you say something about the converse of Rolle's Theorem from this example?

Let $a > 0$ and $f$ be continuous in $[-a, a]$. Suppose that $f'(x)$ exists and $f'(x) \le 1$ for all $x \in (-a, a)$. If $f(a) = a$ and $f(-a) = -a$,then $f(0)$ is: