Question

Find the values of a and b such that the function defined by f(x)=$\{\begin{array}{c}5,ifx\le 2\\ ax+b,if2<x<10\\ 21,ifx\ge 10\end{array}$ is a continuous function.

Answer 1

Here, f(x)= $\{\begin{array}{c}5,ifx\le 2\\ ax+b,if2<x<10\\ 21,ifx\ge 10\end{array}$

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

RHL = $\underset{x\to 2^{+}}{lim}f\left(x\right)=\underset{x\to 2^{+}}{lim}(ax+b)$

Putting x=2+h as $x\to 2^{+}$ when $h\to 0$

$\therefore \underset{h\to 0}{lim}\left[a\right(2+h)+b]=\underset{h\to 0}{lim}(2a+ah+b)=2a+b$

Also, f(2)=5

Since, f(x) is continuous at x=2.

LHL = RHL = f(2)$\Rightarrow$ 2a+b=5 ....(i)

At x = 10, $LHL=\underset{x\to {10}^{-}}{lim}f\left(x\right)=\underset{x\to {10}^{-}}{lim}(ax+b)$

Putting x=10-h as $x\to {10}^{-}$ when $h\to 0$

$\therefore \underset{h\to 0}{lim}\left[a\right(10-h)+b]=\underset{h\to 0}{lim}(10a+ah+b)=10a+b$

RHL = $\underset{x\to {10}^{+}}{lim}f\left(x\right)=\underset{x\to {10}^{+}}{lim}\left(21\right)=21$

Als, f(10)=21

Since, f(x) is continuous at x=10.

LHL=RHL =f(10)$\Rightarrow$ 10a+b=21 ......(ii)

Subtracting Eq. (i) from Eq (ii), we get 8a = 16 $\Rightarrow$ a=2

Put a=2 in Eq. (i), we get 2$\times$2+b=5$\Rightarrow$b=1