If $f(x) = \cos x$ for $0 \le x \le \frac{\pi}{2}$,then the real number $c$ of the Mean Value Theorem is:

Let $f^{\prime}(0)=-3$ and $f^{\prime}(x) \leq 5$ for all real values of $x$. The $f(2)$ can have possible maximum value as

For a polynomial $g(x)$ with real coefficients,let $m_g$ denote the number of distinct real roots of $g(x)$. Suppose $S$ is the set of polynomials with real coefficients defined by $S = \{(x^2-1)^2(a_0+a_1x+a_2x^2+a_3x^3) : a_0, a_1, a_2, a_3 \in \mathbb{R}\}$. For a polynomial $f$,let $f'$ and $f''$ denote its first and second order derivatives,respectively. Then the minimum possible value of $(m_f + m_{f'})$,where $f \in S$,is

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

If $f(x)=|x-2|, x \in[0,4]$ then the Rolle's theorem cannot be applied to the function because

The value of $c$ satisfying the conditions and conclusions of Rolle's theorem for the function $f(x) = x \sqrt{x+6}$ on the interval $x \in [-6, 0]$ is: