Limits, Continuity & Differentiability
Max-Function Continuity and Differentiability
nta_pyq_2023_apr
Grade 12
Question:
Let $[x]$ denote the greatest integer function and $f(x)=\max\{1+x+[x],\ 2+x,\ x+2[x]\}$, $0\leq x\leq 2$, where $m$ is the number of points where $f$ is not continuous and $n$ be the number of points in $(0,2)$ where $f$ is not differentiable. Then $(m+n)^2+2$ is equal to
Step-by-Step Solution
Key Concept: On $[0,1)$: $[x]=0$, so three terms are $x+1,x+2,x$. Max $=x+2$. On $[1,2)$: $[x]=1$, all three equal $x+2$. At $x=2$: $[x]=2$, max$=6\neq f(2^-)=4$.
$m=1,\ n=0$. $(m+n)^2+2=3$.
Correct Answer: 4