Verified answer

For a function to be differentiable, it must be continuous and the gradient must be the same from the left and right side.

For example, a graph shaped like V is continuous but it is not differentiable at the point of the V.

You have a graph that changes from a straight line to a parabola at x = 1. Often that would be a sudden change of shape and would not be differentialble. If the line and the parabola are 'lined up' properly it can be smooth and differentiable.

First of all, the derivative:

f'(x) = 2 for x<= 1

f'(x) = 2bx for x> 1. To make these the same gradient at x= 1 (both sides of the point)

2 = 2b(1)

b = 1

Then consider the graph itself. The line and the parabola must meet at the same point without a jump ( be continuous).

f(1) = 2(1) + a

f(1+) = (1)^2 - 1 = 0

Thus 2 + a = 0

a = -2