If $2 f(x)-3 f\left(\frac{1}{x}\right)=x+1$,then $f^{\prime}(\sqrt{3})$ is equal to

Let $f(x)$ be a polynomial function of the second degree. If $f(1) = f(-1)$ and $a_1, a_2, a_3$ are in $A.P.$,then $f'(a_1), f'(a_2), f'(a_3)$ are in

Let $f: R \rightarrow R$ be a differentiable function such that $|f(x) - f(y)| \leq 2|x - y|^{\frac{3}{2}}$ for all $x, y \in R$. If $f(0) = 1$,then $\int_0^1 f^2(x) dx = $

Let $P$ be a non-zero polynomial such that $P(1+x)=P(1-x)$ for all real $x$ and $P(1)=0$. Let $m$ be the largest integer such that $(x-1)^m$ divides $P(x)$ for all such $P(x)$. Then,$m$ equals

The first derivative of the function $\left[ \cos^{-1}\left( \sin \sqrt{\frac{1+x}{2}} \right) + x^x \right]$ with respect to $x$ at $x = 1$ is

Evaluate: $\frac{d}{d x}\left[e^{\tan ^{-1} x+\cot ^{-1} x}+\frac{x}{2} \sqrt{a^2-x^2}+\frac{a^2}{2} \sin ^{-1} \frac{x}{a}\right]$