#### Question

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

`f(x) = {(5, "," if x <= 2),(ax +b, "," if 2 < x < 10),(21, "," if x >= 10):}`

is a continuous function.

#### Solution

The given function *f *is `f(x) = {(5, "," if x <= 2),(ax +b, "," if 2 < x < 10),(21, "," if x >= 10):}`

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

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

8*a* = 16

⇒ *a* = 2

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

2 × 2 + *b* = 5

⇒ 4 + *b* = 5

⇒ *b* = 1

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