Finding Maxima and Minima Using Derivatives

Classifying Stationary Points:

There are two types of stationary points (turning points):

• A maximum point: the largest value of the function or where the graph reaches its peak .
• A minimum point: the smallest value of the function or where the graph reaches its trough .

A quadratic graph has one turning point – a minimum point or a maximum point. For parabolas (quadratics) the stationary points should be obvious:

• ... a positive parabola ( positive x² term) has a minimum point
• ... a negative parabola ( negative x² term) has a maximum point

Cubic graphs often have two turning points – a minimum point and a maximum point and are also easily recognisable ...

• ... a positive cubic has a maximum point on the left , a minimum on the right
• ... a negative cubic has a minimum on the left , a maximum on the right

Second Derivative Test:

This test is done to show whether a stationary point is a maximum or minimum.

The second derivative ,  \(\frac{d^2y}{dx^2}\), is