Let [t] denote the greatest integer less than or equal to t.

Question

Let $[t]$ denote the greatest integer less than or equal to $t$. Let $f:[0,\infty)	o R$ be a function defined by $f(x)=[\frac{x}{2}+3]-[\sqrt{x}]$. Let $S$ be the set of all points in the interval $[0,8]$ at which $f$ is not continuous. Then $\sum _{}a$ is equal to \_\_\_\_\_\_\_\_\_\_.

Answer: 17

Accepted Answer

$\begin{gathered} f(x) =[\frac{x}{3}+3]-[\sqrt{x}] \ =[\frac{x}{2}]-[\sqrt{x}]+3 \end{gathered}$

Critical points where $f(x)$ might change behaviours when $\frac{x}{2}\in$ integer and $\sqrt{x}\in$ integer

$\Rightarrow$ Critical points,

$\begin{gathered} f(0)=3 \ f({0}^{+})=3 \ f({1}^{-})=3 \ f({1}^{+})=2 \ f({2}^{-})=2 \ f({2}^{+})=3 \ f({3}^{+})=f({3}^{-})=3=f({4}^{+})=f({4}^{-})=f({5}^{+})=f({5}^{-}) \ f({6}^{-})=3 \ f({6}^{+})=4 \ f({7}^{-})=f({7}^{+})=4 \ f({8}^{-})=4 \ f(8)=5 \end{gathered}$

$\begin{gathered} \Rightarrow f(x) \ x=1,2,6,8 \ \Rightarrow \sum _{a\in s}a=1+2+6+8 \ \, =17 \end{gathered}$

Solutions · 1