Applications of Differentiation

Second Derivative Test

What is the Second Derivative Test?

The sign test can often be a more complicated technique in determining the nature of stationary points. It requires students to enter to value, one left and one right of the stationary point. If the curve has two stationary points very close together, students might have to use fractions in the sign test which can become messy.


What's the alternative? The second derivative test works by substituting the x coordinate of the stationary point into the second derivative.

  • If the second derivative is positive, it is a minimum turning point

  • If the second derivative is negative, it is a maximum turning point

  • If the second derivative equals to 0, it is a horizontal point of inflection

Second Derivative Test Summary Table

Worked Example: Second Derivative Test

In this example, we are given the following question.

Step 1: Determine the Stationary Point/s

This question asks for the coordinates of the stationary points, so we'll have to substitute the x values into y to get the y-coordinates of the stationary points.

Step 2: Determine the Second Derivative

Step 3: Substitute the x-Coordinates of the stationary point in the second derivative and determine whether it is positive, negative or 0.

Check your answer using a graphics calculator.

Use the graph feature to check that the stationary points and its nature are correct.


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