Why is the force constant the second derivative of potential energy?

[elemis]

In the attachment below some form of link is made between F(x+xo) and dU/dx

I understand F=-dU/dx but I do not understand the derivation shown to prove that the force constant is equal to the second derivative in the last line.

How do they go about this proof ?

 

[psmt]

As you saw, if you start with U = 1/2 kx^2 and differentiate once, you get the linear force law you want.
Differentiate it a second time.