## Second Derivatives

For certain functions, it is possible to differentiate twice and find the derivative of the derivative. It is often denoted as or . For example, given that then the derivative is and the second derivative is given by .

**How to classify stationary points:**

The second derivative can tell us something about the nature of a stationary point. Suppose that we have found the -coordinates of all of the stationary points by solving . For a minimum, the gradient changes from negative to 0 to positive, i.e. the gradient is increasing. If the gradient is increasing then the gradient of the gradient is positive, i.e. . For a maximum, the gradient changes from positive to 0 to negative, i.e. the gradient is decreasing. If the gradient is decreasing then the gradient of the gradient is negative, i.e. . By putting the -coordinates of the stationary points into , we can classify whether they are minima or maxima by determining whether the second derivative is positive or negative at those -coordinates.

**Example** – *Find and classify the stationary points of .
*

We locate the stationary points by solving . f'(x) is given by

We can solve f'(x)=0 by factorising:

which gives x=1/3 or x=1. The corresponding y coordinates are (don’t be afraid of strange fractions) and . Hence, the stationary points are at (1/3,-131/27) and (1,-5). We can classify the stationary points by substituting the x coordinate of the stationary point into the second derivative and seeing if it is positive or negative. Differentiating a second time gives

. It follows that which is less than 0, and hence (1/3,-131/27) is a MAXIMUM. and (1,-5) is a MINIMUM.

## Stationary Points

Stationary points are the points on a curve where the gradient is 0. This means that at these points the curve is flat. Usually, the gradient of a curve is always changing and so the gradient is only 0 instantaneously (unless the curve is a flat line, in which case, the gradient is always 0).

Recall the graph of . The vertex at the bottom of the curve is a STATIONARY POINT. In this case, there is a stationary point at (0,0).

Now consider the general positive quadratic in the form where . There is a stationary point at the bottom of the curve; this is called a MINIMUM. Now consider a negative quadratic of the form where . There is a stationary point at the top of the curve; this is called a MAXIMUM.

A stationary point can be found by solving , i.e. finding the x coordinate where the gradient is 0. dy/dx is found by differentiating.

**Example** – Find the stationary points on the curve .

Start by solving . i.e. . Factorising gives and so the x coordinates of the stationary points are x=4 and x=1. Substituting these into the y equation gives the coordinates of the stationary points as (4,-28/3) and (1,-1/3).