Find values of x for which the piecewise function g(x): 1) (x+1)^2, x=<0 2) 2x+1, 0<x<33) (4-x)^2, x>=3is differentiable.

Question

I got xer, x cannot equal 3 and 0, but the answer key only has 3. WHy isn't 0 one either since the derivative is different on either sides?

Answer 1

first, g must be continuous
g(0-) = (0+1)^2 = 1
g(0+) = 2*0+1 = 1
g(3-) = 2*3+1 = 7
g(3+) = (4-3)^2 = 1
so, g is continuous at x=0, but not at x=3.

g'(0-) = 2x+2 = 2
g'(0+) = 2
So, g is differentiable at x=0, since the slope is the same from both sides.

So g is differentiable everywhere except at x=3.