Stationary Points for IGCSE Maths
Finding and classifying maximum and minimum points. This subtopic is part of Differentiation in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding stationary points is e
What You Need to Know
Finding and classifying maximum and minimum points. This subtopic is part of Differentiation in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding stationary points is essential for achieving a strong grade in your IGCSE Maths exam.
Understanding Stationary Points
A stationary point is where the gradient equals zero (dy/dx = 0). At a minimum, the curve changes from decreasing to increasing; at a maximum, from increasing to decreasing. To classify, find the second derivative d²y/dx²: if it is positive, the point is a minimum; if negative, it is a maximum; if zero, further investigation is needed. Tested in IGCSE Extended Paper 4, often with a sketch of the curve and finding the nature of turning points.
Step-by-Step Method
- 1
Find dy/dx
Differentiate y = f(x) using the power rule.
- 2
Set dy/dx = 0 and solve
Equate the derivative to zero and solve the resulting equation (often a quadratic). The solutions give the x-coordinates of stationary points.
- 3
Find the y-coordinates
Substitute each x-value back into the original equation y = f(x).
- 4
Find d²y/dx² and substitute
Differentiate dy/dx again to get d²y/dx². Substitute each x-value: positive → minimum, negative → maximum.
- 5
State the coordinates and nature
Write 'minimum at (a, b)' or 'maximum at (a, b)'. For IGCSE, you will always get one minimum and one maximum (or just one stationary point).
Worked Example
Question
y = x³ − 3x² − 9x + 5. Find the coordinates of the stationary points and determine their nature.
Solution
Step 1: dy/dx = 3x² − 6x − 9 Step 2: Set to zero. 3x² − 6x − 9 = 0 x² − 2x − 3 = 0 (x − 3)(x + 1) = 0 x = 3 or x = −1 Step 3: Find y-values. At x=3: y = 27−27−27+5 = −22 At x=−1: y = −1−3+9+5 = 10 Step 4: d²y/dx² = 6x − 6 At x=3: 18−6 = 12 > 0 → MINIMUM At x=−1: −6−6 = −12 < 0 → MAXIMUM Answers: Minimum at (3, −22); Maximum at (−1, 10)
Exam Tips for Stationary Points
- Set dy/dx = 0 to find x — do not set y = 0 (that gives the x-intercepts, which are different).
- Always find the y-coordinates of stationary points — the question asks for coordinates, not just x-values.
- d²y/dx² > 0 → minimum (curve smiles ∪). d²y/dx² < 0 → maximum (curve frowns ∩).
- Show the full solution to dy/dx = 0 — a quadratic equation with both roots is expected.
Practice Questions
Q1: Find the stationary point of y = x² − 8x + 15 and state its nature.
Show hint
dy/dx = 2x−8 = 0 → x=4, y=16−32+15=−1. d²y/dx²=2>0 → minimum at (4,−1).
Q2: y = 2x³ − 3x² − 12x + 1. Find and classify all stationary points.
Show hint
dy/dx=6x²−6x−12=6(x²−x−2)=6(x−2)(x+1)=0 → x=2 or x=−1.
Q3: A curve has dy/dx = 6x² − 6. Find the x-values of the stationary points and determine their nature.
Show hint
6x²−6=0 → x²=1 → x=±1. d²y/dx²=12x. At x=1: 12>0 min. At x=−1: −12<0 max.
Frequently Asked Questions
What is stationary points in IGCSE Maths?
Finding and classifying maximum and minimum points.
Is stationary points in the Core or Extended syllabus?
Stationary Points is part of the Extended only syllabus for IGCSE Mathematics 0580.
How do I revise stationary points effectively?
Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on stationary points rather than trying to cover everything at once.
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