Stationary Points & the First Derivative Test
Learn stationary points and the first derivative test for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-06). Stationary points are where the gradient is zero, and checking the gradient either side reveals whether each one is a peak, a trough or a flat inflection.
You will learn to solve \(f'(x) = 0\), build a gradient table, and classify each point as a local maximum, local minimum or horizontal point of inflection — the essential tool for curve sketching and for finding best or worst values.
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.