The number of polynomials $p: R \rightarrow R$ satisfying $p(0)=0, p(x)>x^2$ for all $x \neq 0$ and $p^{\prime \prime}(0)=\frac{1}{2}$ is

Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ 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 $\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \prime}\right)(x)$ is....................

For the function $f(x) = {e^x},a = 0,b = 1$, the value of $ c$ in mean value theorem will be

Let $f(x) = 8x^3 - 6x^2 - 2x + 1,$ then

A value of $c$ for which conclusion of Mean Value Theorem holds for the function $f\left( x \right) = \log x$ on the interval $[1,3]$ 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 $[0, 1 ].$