Let $f$ be any function continuous on $[\mathrm{a}, \mathrm{b}]$ and twice differentiable on $(a, b) .$ If for all $x \in(a, b)$ $f^{\prime}(\mathrm{x})>0$ and $f^{\prime \prime}(\mathrm{x})<0,$ then for any $\mathrm{c} \in(\mathrm{a}, \mathrm{b})$ $\frac{f(\mathrm{c})-f(\mathrm{a})}{f(\mathrm{b})-f(\mathrm{c})}$ is greater than

For a polynomial $g ( x )$ with real coefficient, let $m _{ g }$ denote the number of distinct real roots of $g ( x )$. Suppose $S$ is the set of polynomials with real coefficient defined by

$S=\left\{\left(x^2-1\right)^2\left(a_0+a_1 x+a_2 x^2+a_3 x^3\right): a_0, a_1, a_2, a_3 \in R\right\} \text {. }$

For a polynomial $f$, let $f^{\prime}$ and $f^{\prime \prime}$ denote its first and second order derivatives, respectively. Then the minimum possible value of $\left(m_f+m_{f^{\prime}}\right)$, where $f \in S$, is. . . . . . . .

Consider a quadratic equation $ax^2 + bx + c = 0,$ where $2a + 3b + 6c = 0$ and let $g(x) = a\frac{{{x^3}}}{3} + b\frac{{{x^2}}}{2} + cx.$

Statement $1:$ The quadratic equation has at least one root in the interval $(0, 1).$

Statement $2:$ The Rolle's theorem is applicable to function $g(x)$ on the interval $[0, 1 ].$

Verify Mean Value Theorem for the function $f(x)=x^{2}$ in the interval $[2,4]$

If $f(x) = \cos x,0 \le x \le {\pi \over 2}$, then the real number $ ‘c’ $ of the mean value theorem is

Let $f(x) = \sqrt {x - 1} + \sqrt {x + 24 - 10\sqrt {x - 1} ;} $ $1 < x < 26$ be real valued function. Then $f\,'(x)$ for $1 < x < 26$ is