Let [x] denote the greatest integer function and f(x)=max1+x+[x],2+x,x+2[x],0 x 2.

Question

Let $[x]$ denote the greatest integer function and

$f(x)=max{1+x+[x],2+x,x+2[x]},0\le x\le 2$. Let $m$ be the number of

points in $[0,2]$, 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 :

Accepted Answer

$$
\begin{aligned}
& \text { Let } g(x)=1+x+[x]=\left\{\[\begin{array}{cc} 1+x ; & x \in[0,1) \ \
2+x ; & x \in[1,2) \ 5 ; & x=2 \end{array}\end{aligned}$$\right. \\
\& \lambda(x)=x+2[x]=\left\{$$\begin{array}{cc} x ; & x \in[0,1) \ x+2 ; & x \in[1,2) \ 6 ; & x=2 \end{array}$$\right. \\
\& r(x)=2+x \\
\& f(x)=\left\{$$\begin{array}{cc} 2+x ; & x \in[0,2) \ 6 ; & x=2 \end{array}$$\right.

\]

$\mathrm{f}(\mathrm{x})$ is discontinuous only at $x=2 \Rightarrow \mathrm{m}=1$

$\mathrm{f}(\mathrm{x})$ is differentiable in $(0,2) \Rightarrow \mathrm{n}=0$

$(m+n)^2+2=3$

Solutions · 1