The second derivative test is used to determine if a given stationary point is a maximum or minimum.
The first step of the second derivative test is to find stationary points.
A stationary point is a point on a curve with a gradient of zero such that or .
Find the stationary points of the curve .
We have Since stationary points occur when , the values of for which are obtained as follows:
Substituting these values of back into the original curve equation gives the full pairs of coordinates:
which implies that the stationary points are and
Note in the example above that the full coordinates were found. When dealing with the second derivative test, only the -coordinate values are needed.
A second derivative is found by differentiating a derivative of a function. In other words, differentiate a function to get its derivative, then differentiate again to get its second derivative.
Notation for second derivatives can be represented as
- , read as "f double-dashed x";
- , read as "d two y by d x squared."
Find the second derivative of .
Differentiate the first time to get the derivative:
Now, differentiate the derivative to get the second derivative: .
Given that , find .
First, find the derivative of .
Now, differentiate to find the second derivative: .
After finding the -values of the stationary points, the second derivative needs to be found. Then, we substitute the -values from the stationary points into the second derivative to perform the test. The test has three outcomes:
- If the second derivative is less than zero, the stationary point is a maximum.
- If the second derivative is greater than zero, the stationary point is a minimum.
- If the second derivative equals zero, the stationary point could be a point of inflection.
The derivative of a curve is found to be . Stationary points on this curve occur when and . Determine if these points are maxima or minima using the second derivative test.
We have .
Now, substitute to determine if it is a maximum or minimum point: .
As , the stationary point when is a maximum.
Do the same for the other value: .
As , the stationary point when is a minimum.