To identify the nature of the points - minimum, maximum or inflexion point, we need to evaluate the second derivative - that is - the derivative of the derivative at the stationary point. This is written

If the point is a maximum, then the gradient of the curve decreases, so at a maximum, . Similarly, if the point is a minimum, . In the case when , the point could be any of the three types of points and the nature has to be determined by a different method. The simplest method is to consider the signs of the function close to the stationary point.

For example, to find the stationary points of the curve , we proceed as follows.

This shows that the curve has two stationary points. To find their nature, we find and evaluate the second derivative at these points

Hence, the first is a maximum and the second is a minimum.

Calculus has lots more applications in geometry in the field known as ``differential geometry'' concerning curvature, solids of revolution, arc length, etc.