Types Of Stationary Points
Learn the types of stationary points for Queensland Year 11 Mathematical Methods (QCAA). Once a curve levels off, it may reach a local maximum, a local minimum, or a stationary point of inflection.
You will learn to determine the nature of each stationary point with the first-derivative sign test, build a sign chart of the gradient, and describe how a curve behaves around each one — essential for accurate curve sketching.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA, Unit 2), the nature of a stationary point — whether it is a local maximum, a local minimum or a stationary point of inflection — is decided with the first-derivative sign test: check the sign of \(f'(x)\) just before and just after the point. This page shows the sign test in full for each type.
Once a stationary point is located by solving \(f'(x)=0\), its nature is how the curve behaves around it. There are three types: a local maximum (a peak), a local minimum (a trough), and a stationary point of inflection (the curve flattens then continues the same way).
The first-derivative sign test classifies the point by the sign of \(f'(x)\) on each side. If \(f'\) changes \(+\rightarrow-\) the point is a maximum; if it changes \(-\rightarrow+\) it is a minimum; if \(f'\) keeps the same sign on both sides (no change) it is a stationary point of inflection.
Let \(x=a\) be a stationary point (\(f'(a)=0\)). Test a value just below and just above \(a\):
How to classify a stationary point by the sign test
- Locate: find \(f'(x)\) and solve \(f'(x)=0\) for the stationary \(x\)-value(s).
- Test each side: choose a value just below and just above each stationary \(x\), substitute into \(f'(x)\), and record the sign.
- Read the nature: \(+\rightarrow-\) is a maximum, \(-\rightarrow+\) is a minimum, and no change is a stationary point of inflection; then find the \(y\)-coordinate from \(f(x)\).
Locate — solve \(f'(x)=0\):
| \(f'(x)\) | \(=\) | \(2x-4\) |
| \(2x-4\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(2\) |
Sign test — test \(x=1\) and \(x=3\):
| \(f'(1)\) | \(=\) | \(2(1)-4=-2<0\) |
| \(f'(3)\) | \(=\) | \(2(3)-4=2>0\) |
The sign changes \(-\rightarrow+\), so \(x=2\) is a local minimum.
\(y\)-coordinate:
| \(f(2)\) | \(=\) | \((2)^2-4(2)+3\) |
| \(=\) | \(-1\) |
Local minimum at \((2,\,-1)\).
Locate — solve \(f'(x)=0\):
| \(f'(x)\) | \(=\) | \(3x^2-3\) |
| \(3x^2-3\) | \(=\) | \(0\) |
| \(3(x-1)(x+1)\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(-1 \ \text{or}\ 1\) |
Sign test — test \(x=-2,\ 0,\ 2\):
| \(f'(-2)\) | \(=\) | \(3(4)-3=9>0\) |
| \(f'(0)\) | \(=\) | \(-3<0\) |
| \(f'(2)\) | \(=\) | \(3(4)-3=9>0\) |
At \(x=-1\): \(+\rightarrow-\), a local maximum. At \(x=1\): \(-\rightarrow+\), a local minimum.
\(y\)-coordinates:
| \(f(-1)\) | \(=\) | \((-1)^3-3(-1)=2\) |
| \(f(1)\) | \(=\) | \((1)^3-3(1)=-2\) |
Local maximum \((-1,\,2)\); local minimum \((1,\,-2)\).
Locate — solve \(f'(x)=0\):
| \(f'(x)\) | \(=\) | \(3x^2-6x+3\) |
| \(3(x-1)^2\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(1\) |
Sign test — test \(x=\tfrac12\) and \(x=\tfrac32\):
| \(f'\!\left(\tfrac12\right)\) | \(=\) | \(3\left(\tfrac12-1\right)^2=\tfrac34>0\) |
| \(f'\!\left(\tfrac32\right)\) | \(=\) | \(3\left(\tfrac32-1\right)^2=\tfrac34>0\) |
The sign is \(+\) on both sides — no change — so this is a stationary point of inflection.
\(y\)-coordinate:
| \(f(1)\) | \(=\) | \((1)^3-3(1)^2+3(1)+1\) |
| \(=\) | \(2\) |
Stationary point of inflection at \((1,\,2)\).
Locate — solve \(f'(x)=0\):
| \(f'(x)\) | \(=\) | \(4x^3-12x^2\) |
| \(4x^2(x-3)\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(0 \ \text{or}\ 3\) |
Sign test — test \(x=-1,\ 1,\ 4\):
| \(f'(-1)\) | \(=\) | \(4(-1)(-4)=-16<0\) |
| \(f'(1)\) | \(=\) | \(4(1)(-2)=-8<0\) |
| \(f'(4)\) | \(=\) | \(4(16)(1)=64>0\) |
At \(x=0\): \(-\rightarrow-\), no change, a stationary inflection. At \(x=3\): \(-\rightarrow+\), a local minimum.
\(y\)-coordinates:
| \(f(0)\) | \(=\) | \(0\) |
| \(f(3)\) | \(=\) | \((3)^4-4(3)^3=81-108=-27\) |
Stationary inflection \((0,\,0)\); local minimum \((3,\,-27)\).
Common pitfalls
Frequently asked questions
How do you tell if a stationary point is a maximum or minimum?
Use the first-derivative sign test: if \(f'\) changes from \(+\) to \(-\) it is a local maximum, and from \(-\) to \(+\) it is a local minimum.
What is a stationary point of inflection?
A stationary point where \(f'(x)=0\) but the sign of \(f'\) does not change — the curve flattens then keeps going the same way.
How do I run a first-derivative sign test?
Pick a value just below and just above the stationary \(x\), substitute each into \(f'(x)\), and note the signs; the pattern of change gives the nature.
Can I use the second derivative instead?
Not in Year 11 Methods — the syllabus classifies stationary points with the first-derivative sign test. The second-derivative test is Year 12.
Does f'(x)=0 tell me the type of point?
No. It only locates the stationary point. You still need the sign test to decide maximum, minimum or inflection.
What if the two test signs are the same?
No sign change means the point is a stationary point of inflection, not a maximum or a minimum.