Radford Mathematics IB Mathematics resources • AA HL

IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus

Stationary Points and Derivative Tests

Classify stationary points reliably with first- and second-derivative tests.

AA HL · SL 5.8

Learning goal

Find and classify stationary points using derivative sign changes and the second derivative test.

Syllabus link

AA HL: SL 5.8 · Topic 5 Calculus.

Big idea

A stationary point is found from the first derivative; its nature is determined by nearby behaviour or by a decisive second derivative.

Key relationship

\(f\'(a)=0\)

1

Core idea

A stationary point is found from the first derivative; its nature is determined by nearby behaviour or by a decisive second derivative.

Find and classify stationary points using derivative sign changes and the second derivative test.

2

Key results and method

\[f\'(a)=0\]
\[f\'\'(a)>0\Rightarrow\text{local minimum}\]
\[f\'\'(a)<0\Rightarrow\text{local maximum}\]
  1. Solve \(f'(x)=0\).
  2. Evaluate \(f\) to get point coordinates.
  3. Use the second derivative test when non-zero.
  4. If the second derivative test is inconclusive, use a first-derivative sign table.
3

Worked example

Question

Classify the stationary points of \(f(x)=x^3-3x\).

Solution
  1. \(f\'(x)=3x^2-3=0\Rightarrow x=\pm1\).
  2. \(f\'\'(x)=6x\).
  3. At \(x=-1\), \(f\'\'<0\); at \(x=1\), \(f\'\'>0\).
Answer: Maximum at \((-1,2)\), minimum at \((1,-2)\).
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Practice

Try these questions before moving on.

  1. Classify a stationary point where \(f''(a)=0\) using a sign table.
  2. Sketch the derivative sign pattern of a local minimum.
  3. Find all stationary points of a quartic and classify them.

Search the worksheet library for more practice →

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Tutorials and premium resources

▶ When the second derivative test fails
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