Question

If $a$ and $b$ such that the function
$f\left(x\right)=\{5,ifx\le 2$
$ax+b,if2<x<10$
$21,ifx\ge 10$
is a continuous function.Then $a=\frac{8}{m}$.Find $m$

Answer 1

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

$\therefore$ fx) is a continuous function.

At $x=2$

L.H.L(Left Hand Limit)

$\underset{x\to 2^{-}}{lim}f\left(x\right)=5$

R.H.L(Right Hand Limit)

$\underset{x\to 2^{+}}{lim}f\left(x\right)=2a+b$-Equation 1

At $x=10$

L.H.L(Left Hand Limit)

$\underset{x\to {10}^{-}}{lim}f\left(x\right)=10a+b$

R.H.L(Right Hand Limit)

$limx\to {10}^{+}=21$

$\therefore L.H.L=R.H.L\Rightarrow 10a+b=21$-Equation 2

On substraction, we get

$8a=16\therefore a=2$

$a=\frac{8}{m}\therefore m=4$