Question

Find the values of $a$ and $b$ such that the function defined by $f\left(x\right)=\{\begin{array}{ccc}5,& if& x\le 2\\ ax+b,& if& 2<x<10\\ 21,& if& x\ge 10\end{array}$ is a continuous function.

Answer 1

$f\left(x\right)=\{\begin{array}{ccc}5,& if& x\le 2\\ ax+b,& if& 2<x<10\\ 21,& if& x\ge 10\end{array}$

If the function $f\left(x\right)$ is a continuous function, then

$f\left(2\right)=f\left(2^{+}\right)$ and $f\left({10}^{-}\right)=f\left(10\right)$

$I:f\left(2\right)=f\left(2^{+}\right)$

$5=2a+b$ ... (i)

$II:f\left({10}^{-}\right)=f\left(10\right)$

$10a+b=21$ ...(ii)

from (i) and (ii), we get

$a=2,b=1$