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Principle deep dive

AM-GM / mean inequalities (incl. x + 1/x ≥ 2)

Whenever you need a minimum of a sum or maximum of a product under a constraint, AM-GM is the lever. Spans Sequence & Series, Application of Derivatives, Trig Identities and Logarithms; questions disguise the inequality across chapter lines.

questions in the bank
31
tagged HARD
16%
chapter spread
6
worked examples below
4

When to reach for it

You need a minimum of a sum or a maximum of a product, given a constraint on the other.

Why this principle matters

AM-GM is the lever most students reach for last. They start with calculus, set f'(x) = 0, lose to messy algebra, and then realise the answer was a one-line bound. The principle is simple: for positive reals, the arithmetic mean is at least the geometric mean, with equality when all values are equal.

The bank disguises this across 10 chapters. Sequence & Series asks for AP/GP relations. Application of Derivatives wraps it in 'find the minimum'. Trigonometric Identities asks for the max of sin·cos. Probability frames it as 'find x such that x + 1/x > 2'. The chapter changes; the inequality doesn't.

Internalise three patterns: (1) for positive x, x + 1/x ≥ 2 with equality at x = 1, (2) for fixed product xy, the sum x + y is minimised when x = y, (3) for fixed sum x + y, the product xy is maximised when x = y. From these you can derive most of the bank's AM-GM questions in under 60 seconds each.

4 worked examples from the bank

Each example demonstrates the principle on a real past-year question. Click to reveal the answer, then the solution.

Example 1Application of DerivativesEASY
If xy=4225xy=4225 where x,yx, y are natural numbers, then what is the minimum value of x+yx+y?

[Q79 · Apr · 2022]

Example 2Application of DerivativesEASY
If x+y=20x+y=20 and P=xyP=xy, then what is the maximum value of PP?

[Q87 · Apr · 2021]

Example 3Trigonometric IdentitiesEASY
If A and B are acute angles such that 2A+2B=π2A+2B=\pi, then what is the maximum value of sin⁡A⋅sin⁡B\sin A\cdot\sin B?

[Q78 · Apr · 2026]

Example 4Application of DerivativesHARD
Under which one of the following conditions does the function f(x)=(psec⁡x)2+(qcsc⁡x)2f(x) = (p\sec x)^2+(q\csc x)^2 attain minimum value?

[Q91 · Sep · 2022]

Variants to recognise

Same principle, different surfaces. Pattern-match these on test day.

  • x + 1/x ≥ 2

    For positive x. Generalises to xⁿ + 1/xⁿ ≥ 2 for any positive n.

  • Cauchy-Schwarz / power mean inequality

    Generalisation: AM of pᵗʰ powers ≥ AM of qᵗʰ powers for p > q. NDA rarely needs this; AM-GM suffices.

  • Three-variable AM-GM

    If abc = k, then a + b + c ≥ 3·k^(1/3). Equality at a = b = c = k^(1/3).

  • AM ≥ GM ≥ HM chain

    For positive reals, AM ≥ GM ≥ HM with all three equal iff all values equal. Two more inequalities for free.

Drill every am-gm / mean inequalities (incl. x + 1/x ≥ 2) question

31 questions from the bank — paginated, with cart and Word-export support.

Drill the 31 questions

Related principles

Often combined with this one — drill these next if you found the examples above tractable.