Positive terms with a fixed product?
Try AM–GM.
H3 MATHEMATICS · INEQUALITIES
Match the expression.
Then prove the bound.
A proof-focused guide for Singapore A-Level H3 Mathematics 9820. Learn when AM–GM or Cauchy–Schwarz fits, and treat the equality condition as part of the solution.
- Proof structure
- Equality conditions
- Technique selection
Built around 9820 H3 MathematicsAM–GM, Cauchy–Schwarz, triangle inequality and proof
Proof, not pattern matchingState conditions, justify each step and identify equality
THE CENTRAL HABIT
See the target shape first
AM–GM is natural for positive quantities when a product is controlled. Cauchy–Schwarz is powerful when sums of products or reciprocal pairs appear.
A complete proof also states why the inequality applies and when equality holds.
THE SELECTOR
Choose the structure that matches the expression
Sums of products?
Try Cauchy–Schwarz.
Need the sharp bound?
Track the equality condition from the chosen theorem and confirm it is allowed by the problem.
PROOF EXAMPLE
Pair each term with its reciprocal.
For positive real numbers x and y, prove that (x + y)(1/x + 1/y) ≥ 4, and determine when equality holds.
EXAM HABIT
Write the equality condition immediately after applying an inequality.
- 1
Choose two matching pairs
Apply Cauchy–Schwarz to (√x, √y) and (1/√x, 1/√y).
- 2
Apply the inequality
- 3
State the equality condition
Equality holds when the two vectors are proportional, which gives x = y.
ConclusionThe expression is at least 4, with equality exactly when x = y.
COMMON MISTAKES
What weakens an otherwise good proof
Ignoring positivity
AM–GM and reciprocal substitutions require their conditions to be stated and satisfied.
Using a true but weak bound
The method must reach the target constant, not merely produce some inequality.
Dropping equality
A sharp-bound question is incomplete without when the bound is attained.
Working backwards only
Use exploration privately, then present a forward chain in which every implication is valid.
PRACTISE WITH FEEDBACK
Make every implication defensible.
Ask Integrand to critique a handwritten proof, identify a missing condition or suggest a first step without revealing the whole argument.
Try Integrand- 1
Observe
Rewrite the target into a familiar structure.
- 2
Justify
Name the inequality and verify its conditions.
- 3
Finish
State the bound and equality case precisely.
QUICK QUESTIONS
H3 inequalities FAQ
How do I decide between AM–GM and Cauchy–Schwarz?
AM–GM often suits positive terms with a useful product. Cauchy–Schwarz often suits sums of products, squares or reciprocal pairs. Start from the target form rather than the theorem name.
Why is the equality condition important?
It shows that the bound is attainable and identifies the exact case in which it is attained.
Are these inequalities in the current H3 syllabus?
Yes. AM–GM, Cauchy–Schwarz and the triangle inequality are explicitly listed as additional inequalities in Singapore A-Level H3 Mathematics syllabus 9820.
READY TO PRACTISE?