Find the values of a and b that make f continuous everywhere.

Question

x² − 4 / x − 2 if x < 2

f(x) = { ax² − bx + 3 if 2 ≤ x < 3

4x − a + b if x ≥ 3

a=______

b=______

Answer 1

The polynomials are all continuous in the domains given.

The limit of the first expression as x-> 2 is 4.

This means 4a-2b+3=4 => 4a - 2b =1

The last 2 expressions need to be equal at x=3:

9a-3b+3=12-a+b => 10a - 4b =9

Multiply the 1st equation by 2 and subtract: 2a=7 => a=7/2

Plug this into either equation to get b=13/2