Find the values of a and b such that the function defined by

Solution:

The given function $f$ is $f(x)= \begin{cases}5, & \text { if } x \leq 2 \\ a x+b, & \text { if } 2

It is evident that the given function f is defined at all points of the real line.

If f is a continuous function, then f is continuous at all real numbers.

In particular, f is continuous at x = 2 and x = 10

Since f is continuous at x = 2, we obtain

$\lim _{x \rightarrow 2^{+}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)=f(2)$

$\Rightarrow \lim _{x \rightarrow 2^{-}}(5)=\lim _{x \rightarrow 2^{+}}(a x+b)=5$

$\Rightarrow 5=2 a+b=5$

$\Rightarrow 2 a+b=5$   ....(1)

Since f is continuous at x = 10, we obtain

$\lim _{x \rightarrow 10^{-}} f(x)=\lim _{x \rightarrow 10^{+}} f(x)=f(10)$

$\Rightarrow \lim _{x \rightarrow 10^{-}}(a x+b)=\lim _{x \rightarrow 10^{+}}(21)=21$

$\Rightarrow 10 a+b=21=21$

$\Rightarrow 10 a+b=21$  ....(2)

On subtracting equation (1) from equation (2), we obtain

8a = 16

$\Rightarrow a=2$

By putting a = 2 in equation (1), we obtain

2 × 2 + b = 5

$\Rightarrow 4+b=5$

$\Rightarrow b=1$

Therefore, the values of a and b for which f is a continuous function are 2 and 1 respectively.