For what values of a and b is the function continuous and differentiable?

Question

g(x) = { x^2-a , x≤ 2

bx-8 , x>2

Answer 1

First, for the piecewise function to be continuous, the two pieces have to equal each other at x = 2.

So x^2-a = bx-8 at x = 2

(2)^2-a = b(2)-8

4-a = 2b-8

Second, for the piecewise function to be differentiable, the derivatives of the two pieces have to equal each other at x = 2.

g ' (x) = 2x, x <= 2 and b, x > 2

So 2x = b at x = 2

2(2) = b

4=b

Now plug 4 in for b in our first equation to find a.

4-a=2(4)-8

4-a=8-8

4-a=0

4=a

So a = 4 and b = 4