For the differentiable function $f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$, let $3 f(x)+2 f\left(\frac{1}{x}\right)=\frac{1}{x}-10$, then $\left|f(3)+f^{\prime}\left(\frac{1}{4}\right)\right|$ is equal to

If $2 x^y+3 y^x=20$, then $\frac{d y}{d x}$ at $(2,2)$ is equal to

If a function f satisfies f(m+n)=f(m)+f(n) for all m,n∈N and f(1)=1, then the largest natural number λ such that $∑_(k=1)^2022 f(λ+k)≤(2022)^2 $ is equal to

Let $f(x)=a x^3+b x^2+e x+41$ be such that $f(1)=40, f^{\prime}(1)=2$ and $f^{\prime \prime}(1)=4$. Then $a^2+b^2+c^2$ is equal to :

Let $f(\mathrm{x})=\mathrm{x}^5+2 \mathrm{e}^{\mathrm{x} / 4}$ for all $\mathrm{x} \in \mathrm{R}$. Consider a function $g(x)$ such that (gof) $(x)=x$ for all $x \in R$. Then the...

If $f(x)=\left\{\begin{array}{cc}x^3 \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x=0\end{array}\right.$, then

$f(x)=\left|\begin{array}{ccc}3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2\end{array}\right|$ for all $x \in \mathbb{R}$, then $2 f(0)+f^{\prime}(0)$ is equal to

If $...

Suppose $f(x)=\frac{\left(2^x+2^{-x}\right) \tan x \sqrt{\tan ^{-1}\left(x^2-x+1\right)}}{\left(7 x^2+3 x+1\right)^3}$ Then the value of f^' (0) is equal to

Let for a differentiable function $f:(0, \infty) \rightarrow R$,
$f(x)-f(y) \geq \log _{\varepsilon}\left(\frac{x}{y}\right)+x-y, \forall x, y \in(0, \infty).$
Then $\sum_{n=1}^{20} f^{\prime}\left(\frac{1}{n^2}\right)$ is equal to