What is the second derivative used for?

First of all, "second derivative", d²y/dx², is what you get when you differentiate the first derivative (dy/dx).

The second derivative can be used as an easier way of determining the stationary points of a curve.

A stationary point on a curve can be a maximum point, a minimum point or a point of inflection. Those occur when dy/dx = 0. Once you have established where there is a stationary point, the type of stationary point (maximum, minimum or point of inflection) can be determined using the second derivative.

Thus,

If d²y/dx² (second derivative of y in terms of x)  is positive, then it is a minimum point

If d²y/dx² is negative, then it is a maximum point

If d²y/dx² is zero, then it could be a maximum, minimum or point of inflection.

If d²y/dx² is 0, you must test the values of dy/dx (first derivative) either side of the stationary point, as before in the stationary points section.

If dy/dx is possitive before and negative after the stationary point then the last is a maximum. 

If dy/dx is negative before and possitive after the stationary point then the last is a minimum.