JEE Main 2021 (Online) 27th July Morning ShiftQuestion 1
Let $f:[0,3]\to\mathbf{R}$ be defined by $f(x)=\min\{x-[x],1+[x]-x\}$ where $[x]$ is the greatest integer less than or equal to $x$. Let $P$ denote the set containing all $x\in[0,3]$ where $f$ is discontinuous, and $Q$ denote the set containing all $x\in(0,3)$ where $f$ is not differentiable. Then the sum of number of elements in $P$ and $Q$ is equal to ________.
Differentiation and its Application
Answer$5$
Solution
Concept. $f(x)$ is the distance from $x$ to the nearest integer: a row of tents, with peaks at half-integers and valleys at integers. Formulas used. (1) On $[n,n+1)$, with $t=x-n$: $f=\min\{t,1-t\}$Step 1 - Continuity. By (1), $f\to0$ at each integer from both sides, and $f(n)=0$. So $f$ is continuous on $[0,3]$: $|P|=0$. Step 2 - Corners in $(0,3)$. Valleys at $x=1,2$ (slopes $-1$ and $1$). Peaks at $x=\frac12,\frac32,\frac52$ (slopes $1$ and $-1$). So $|Q|=5$. Step 3 - Sum. $0+5=5$. Answer: 5