Find stationary points of a curve
Locate points where the derivative is zero, solve for their coordinates and distinguish stationary points from derivative failures.
Section 14 · Calculus
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the tangent gradient at a stationary point?
What you will learn
- Find stationary inputs by solving f′(x) = 0
- Calculate the coordinates of each stationary point
- Recognise that stationary points can be maxima, minima or neither
A stationary point has a horizontal tangent
At a differentiable stationary point, f′(x) = 0. Solve this equation to find candidate x-values, then substitute each into f(x) to obtain the full coordinates.
A zero derivative does not by itself say whether the point is a maximum or minimum. A curve can flatten and continue increasing or decreasing; use a first- or second-derivative test to classify it.
Keep the equation and point distinct
The equation f′(x) = 0 gives stationary inputs, while (x, f(x)) gives the stationary points on the original curve.
A point where a derivative is undefined is not automatically a stationary point under this test. Check differentiability and the domain before classifying candidates.
Pause and explain
If f′(x) = 0 at x = 2, what else must you calculate to give the point?
Worked example
Find the stationary points of f(x) = x³ − 3x² − 9x + 1.
Show the worked solution
- Differentiate: f′(x) = 3x² − 6x − 9 = 3(x − 3)(x + 1).
- Set f′(x) = 0 to get x = 3 or x = −1.
- Substitute into f: f(3) = −26 and f(−1) = 6.
Answer Stationary points (−1, 6) and (3, −26); classify them separately.
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the stationary coordinates of f(x) = x³ − 6x² + 9x + 2.
Give me a hint
Factor the first derivative, then substitute both inputs into f.
Compare my reasoning
- f′(x) = 3x² − 12x + 9 = 3(x − 1)(x − 3).
- Stationary inputs are x = 1 and x = 3.
- f(1) = 6 and f(3) = 2.
Stationary points (1, 6) and (3, 2).
Look for these in your work
- Derivative factored
- Both inputs found
- Coordinates evaluated on the original function
Find the stationary points of f(x) = x³ − 3x.
Give me a hint
Solve the derivative equation and evaluate the original cubic at each root.
Compare my reasoning
- f′(x) = 3x² − 3 = 3(x − 1)(x + 1).
- The stationary inputs are x = −1 and x = 1.
- f(−1) = 2 and f(1) = −2.
Stationary points (−1, 2) and (1, −2).
Look for these in your work
- Derivative solved
- Coordinates calculated
- Point classification left to a test
A particle's displacement is s(t) = t³ − 6t² + 9t. Find the times when its velocity is zero and the corresponding displacements.
Give me a hint
Velocity is ds/dt; solve v(t) = 0 and substitute into s.
Compare my reasoning
- v(t) = 3t² − 12t + 9 = 3(t − 1)(t − 3).
- Velocity is zero at t = 1 and t = 3 seconds.
- s(1) = 4 and s(3) = 0 metres.
At t = 1 s, s = 4 m; at t = 3 s, s = 0 m.
Look for these in your work
- Velocity derived
- Both times found
- Positions evaluated
Common mistakes
- Solving f(x) = 0 instead of f′(x) = 0
- Reporting stationary x-values without their y-coordinates
- Assuming every stationary point is a maximum or minimum
- Including a candidate outside the function's domain
What can you explain now?
What equation is solved to find stationary x-values?
Compare with the explanation
f′(x) = 0
A differentiable stationary point has a horizontal tangent.
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Make it stick
How do you move from f′(x) = 0 to the coordinates of each stationary point?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.