JEE Main 2021 (Online) 3rd August Evening ShiftQuestion 1
Let $f:\mathbf{R}\to\mathbf{R}$ be a differentiable function such that $f(0)=2$. If $\left|\frac{df}{dx}\right|\le3$, for all $x\in\mathbf{R}$, then $f(1)$ lies in the interval
Differentiation and its Application
Solution
Concept. The mean value theorem turns the bound on $f'$ into a bound on $f(1)-f(0)$. Formulas used. (1) MVT: $f(1)-f(0)=f'(c)$ for some $c\in(0,1)$Step 1 - Bound. By (1), $|f(1)-2|=|f'(c)|\le3$. Step 2 - Interval. $-1\le f(1)\le5$. Answer: (C)