Stationary Points & the First Derivative Test
Stationary points occur where \(f'(x)=0\); the first derivative test classifies each as a maximum, minimum or horizontal point of inflection.
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Applications of calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
Theory
Stationary points are where \(f'(x)=0\), classified by how \(f'\) changes sign. This Year 12 Mathematics Advanced topic (MAV-12-06) finds and classifies them.
Solve \(f'(x)=0\), then test the sign of \(f'\) either side: \(+\) then \(-\) is a maximum; \(-\) then \(+\) is a minimum; the same sign both sides is a horizontal point of inflection. (Or use \(f''\): \(<0\) max, \(>0\) min.)
Method
- Solve \(f'(x)=0\).
- Classify by a sign change of \(f'\) (or the sign of \(f''\)).
- Find the \(y\)-values by substituting into \(f\).
| \(2x-6\) | \(=\) | \(0\Rightarrow x=3\) |
| \(f''\) | \(=\) | \(2>0\ (\text{min})\) |
| \(f(3)\) | \(=\) | \(-4\) |
| \(3x^{2}-3\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(\pm 1\) |
| \(f''(-1)\) | \(=\) | \(-6<0\ (\text{max})\) |
| \(f''(1)\) | \(=\) | \(6>0\ (\text{min})\) |
\(f'=3x^{2}=0\Rightarrow x=0\); \(f'\) is \(+\) both sides, so a horizontal point of inflection at \((0,0)\).
Common pitfalls
Frequently asked questions
What is a stationary point?
A point where the derivative is zero, so the curve is momentarily flat.
How do you classify a stationary point?
Check the sign of the first derivative either side (plus then minus is a max), or use the second derivative (negative is a max, positive is a min).
What is a horizontal point of inflection?
A stationary point where the derivative does not change sign, so it is neither a maximum nor a minimum.
What is the first derivative test?
Testing the sign of f prime just before and after a stationary point to decide if it is a maximum, minimum or inflection.