JEE Main 2021 (Online) 18th March Evening ShiftQuestion 1
Let $g(x)=\int_{0}^{x}f(t)\,dt$, where $f$ is continuous function in $[0,3]$ such that $\frac{1}{3}\le f(t)\le1$ for all $t\in[0,1]$ and $0\le f(t)\le\frac{1}{2}$ for all $t\in(1,3]$. The largest possible interval in which $g(3)$ lies is :
Definite Integration
Solution
Concept. Bound $\int_0^1f$ and $\int_1^3f$ separately using the given bounds on $f$. Formulas used. (1) $m\le f\le M$ on an interval of length $L$ $\Rightarrow$ $mL\le\int f\le ML$Step 1 - $[0,1]$. By (1), $\frac13\le\int_0^1f\le1$. Step 2 - $(1,3]$. Length $2$: $0\le\int_1^3f\le1$. Step 3 - Add. $g(3)\in\big[\frac13,2\big]$. Answer: (A)