IB Mathematics: Applications and Interpretation SL — Topic 5 Calculus
Stationary Points
Use technology to generate the derivative, solve f′(x)=0 and classify local maxima and minima.
Learning goal
Find stationary points and interpret local maxima/minima, including with GDC-generated derivative graphs.
Syllabus link
AI SL: SL 5.6 · Topic 5 Calculus.
Big idea
A stationary point occurs where the gradient is zero. Classification depends on behaviour around the point, not on the equation f′(x)=0 alone.
Key relationship
\(f\'(x)=0\)
Core idea
A stationary point occurs where the gradient is zero. Classification depends on behaviour around the point, not on the equation f′(x)=0 alone.
Find stationary points and interpret local maxima/minima, including with GDC-generated derivative graphs.
Key results and method
- Graph \(f\) and/or generate \(f'\) on the GDC.
- Solve \(f'(x)=0\).
- Use the sign of the derivative or the graph to classify the point.
- Check endpoints separately when a greatest/least value on a closed interval is requested.
Worked example
For \(f(x)=x^3-3x\), classify the stationary points.
- \(f\'(x)=3x^2-3=0\Rightarrow x=\pm1\).
- \(f(-1)=2\), \(f(1)=-2\).
- The derivative changes \(+\to-\) at \(-1\) and \(-\to+\) at \(1\).
Practice
Try these questions before moving on.
- Use a GDC to solve \(f'(x)=0\) for a non-polynomial function.
- Explain why a local minimum need not be the least value on a restricted domain.
- Sketch a possible derivative graph for a function with two stationary points.
Tutorials and premium resources
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