Question

Find the values of a and b so that the function f given by
$f\left(x\right)=\left\{\begin{array}{cc}1,& ifx\le 3\\ ax+b,& if3<x<5\\ 7,& ifx\ge 5\end{array}\right.$is continuous at x = 3 and x = 5.

Answer 1

Given: $f\left(x\right)=\left\{\begin{array}{c}1,\mathrm{if}\mathrm{x}\le 3\\ ax+b,\mathrm{if}3<x<5\\ 7,\mathrm{if}\mathrm{x}\ge 5\end{array}\right.$

We have
(LHL at x = 3) = $\underset{x\to 3^{-}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(3-h\right)$$=\underset{h\to 0}{\lim }\left(1\right)=1$

(RHL at x = 3) = $\underset{x\to 3^{+}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(3+h\right)$$=\underset{h\to 0}{\lim }a\left(3+h\right)+b=3a+b$

(LHL at x = 5) = $\underset{x\to 5^{-}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(5-h\right)$$=\underset{h\to 0}{\lim }\left(a\left(5-h\right)+b\right)=5a+b$

(RHL at x = 5) = $\underset{x\to 5^{+}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(5+h\right)$$=\underset{h\to 0}{\lim }7=7$

If f(x) is continuous at x = 3 and 5, then

∴ ​$\underset{x\to 3^{-}}{\lim }f\left(x\right)=\underset{x\to 3^{+}}{\lim }f\left(x\right)\mathrm{and}\underset{x\to 5^{-}}{\lim }f\left(x\right)=\underset{x\to 5^{+}}{\lim }f\left(x\right)$
$\Rightarrow 1=3a+b...\left(1\right)\mathrm{and}5a+b=7...\left(2\right)$

On solving eqs. (1) and (2), we get
$a=3\mathrm{and}b=-8$