Question

If the following function is continuous at $x=3$ and $x=5$, then find $a$ and $b$.
$f\left(x\right)=\{\begin{array}{c}1,\text{if}x\le 3\\ ax+b,\text{if}3<x<5\\ 7,\text{if}x\ge 5\end{array}$

Answer 1

We have,

$f\left(x\right)=\{\begin{array}{c}1,ifx\le 3\\ ax+b,if3<x<5\\ 7,ifx\ge 5\end{array}$

Since function is continuous at $x=3$

${lim}_{x\to 3^{-}}f\left(x\right)={lim}_{x\to 3^{+}}f\left(x\right)$

$1={lim}_{x\to 3^{+}}(ax+b)$

$3a+b=1$ $..............\left(1\right)$

Since, function is continuous at $x=5$

${lim}_{x\to 5^{-}}f\left(x\right)={lim}_{x\to 5^{+}}f\left(x\right)$

${lim}_{x\to 5^{+}}(ax+b)=7$

$5a+b=7$ $..............\left(2\right)$

On solving equation $\left(1\right)$ and $\left(2\right)$, we get

$a=3,b=-8$

Hence, this is the answer.