CalculusAdditional / Further Maths

Classifying stationary points

Use the second derivative to classify a stationary point.

  • First solve .
  • If , there is a local minimum. If , there is a local maximum.
  • If , this test is inconclusive; use another method.

Example 1

Find the stationary point and classify it using the second derivative.

Differentiate.Set the gradient equal to zero.Find the stationary value.Differentiate again.Positive: local minimum.

Example 2

Find the stationary point and classify it using the second derivative.

Differentiate.Set the gradient equal to zero.Find the stationary value.Differentiate again.Negative: local maximum.

Remember
The questions here give a non-zero second derivative at the stationary point.

Your turn

Question 1

Find the stationary point and classify it using the second derivative.

Check answer
Differentiate.Set the gradient equal to zero.Find the stationary value.Differentiate again.Positive: local minimum.
Question 2

Find the stationary point and classify it using the second derivative.

Check answer
Differentiate.Set the gradient equal to zero.Find the stationary value.Differentiate again.Negative: local maximum.
Question 3

Find the stationary point and classify it using the second derivative.

Check answer
Differentiate.Set the gradient equal to zero.Find the stationary value.Differentiate again.Negative: local maximum.
Question 4

Find the stationary point and classify it using the second derivative.

Check answer
Differentiate.Set the gradient equal to zero.Find the stationary value.Differentiate again.Positive: local minimum.
Question 5

Find the stationary point and classify it using the second derivative.

Check answer
Differentiate.Set the gradient equal to zero.Find the stationary value.Differentiate again.Positive: local minimum.
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