JEE Main 2020 (Online) 4th September Morning ShiftQuestion 1
Suppose a differentiable function $f(x)$ satisfies the identity $f(x+y)=f(x)+f(y)+xy^2+x^2y$, for all real $x$ and $y$. If $\lim_{x\to0}\frac{f(x)}x=1$, then $f'(3)$ is equal to ________.
Differentiation and its Application
Answer$10$
Solution
Concept. Use the functional equation inside the definition of $f'(x)$. Formulas used. (1) $f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}$Step 1 - Substitute. $f(x+h)-f(x)=f(h)+xh^2+x^2h$, so by (1) $$f'(x)=\lim_{h\to0}\left(\frac{f(h)}{h}+xh+x^2\right)=1+x^2.$$ Step 2 - Value. $f'(3)=10$. Answer: 10