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Concept 03 / 07Sequences and Series
The AM-GM Inequality and Its Use in Maximum-Minimum Problems
The AM-GM inequality for n positive numbers and its equality condition, splitting terms to make the product constant, weighted AM-GM, the power-mean and Cauchy-Schwarz inequalities, and common pitfalls when the equality case is unreachable.
For positive real numbers a1,a2,…,an,
na1+a2+⋯+an≥na1a2⋯an,
with equality if and only if all the ai are equal. It also extends to the harmonic mean: AM≥GM≥HM, where HM=∑1/ain.
How it is used. The inequality converts between sums and products. If the product of positive quantities is fixed, their sum is minimised when they are equal; if their sum is fixed, their product is maximised when they are equal. Every application follows one of these two patterns, and the art is in choosing the quantities so that the product (or sum) really is constant.
Pattern 1: minimise a sum with a constant product. For x>0, x+x1≥2x⋅x1=2, with equality at x=1. For x+x24, splitting x into two equal halves makes the product constant: 2x+2x+x24≥332x⋅2x⋅x24=3, with equality when 2x=x24, that is, x=2.
Pattern 2: maximise a product with a constant sum. If x+y+z=12 with positive terms, then xyz≤43=64. To maximise x2y subject to x+y=9, split x into two halves: 2x+2x+y=9, so 2x⋅2x⋅y≤27 and x2y≤108, with equality when 2x=y=3, that is, x=6, y=3.
Weighted AM-GM: for positive weights wi with sum W, W∑wiai≥(∏aiwi)1/W. Splitting a term into equal parts, as above, is weighted AM-GM with integer weights.
The equality check is essential. AM-GM gives a bound, but the bound is the minimum or maximum only if equality can actually occur within the constraints. For example, on the interval x≥3, the inequality x+x1≥2 is true but useless, because equality needs x=1. There the function is increasing and the minimum is 3+31=310 at x=3. When the equality case is outside the allowed region, switch to monotonicity or calculus.
Related inequalities.
Power means: for positive numbers, (n∑ai2)1/2≥n∑ai, so the root-mean-square is at least the arithmetic mean. In particular a2+b2≥2(a+b)2.
Cauchy-Schwarz: (∑aibi)2≤(∑ai2)(∑bi2), with equality when the sequences are proportional. It is the algebraic form of ∣u⋅v∣≤∣u∣∣v∣.
From AM-HM: (a1+⋯+an)(a11+⋯+an1)≥n2.
The mth power mean inequality: n∑aim≥(n∑ai)m for m>1 or m<0, and the reverse for 0<m<1.
Key Takeaways:
- n∑ai≥n∏ai for positive numbers; equality iff all equal.
- Constant product ⇒ minimum sum at equality; constant sum ⇒ maximum product at equality.
- Split terms into equal parts so that the product or sum becomes constant.
- Always confirm that the equality case lies within the allowed values.
- (∑ai)(∑ai1)≥n2; a2+b2≥2(a+b)2.
- Cauchy-Schwarz: (∑aibi)2≤∑ai2∑bi2.
- Pitfall: applying AM-GM to quantities that can be negative.
Brain-Teaser 1: Find the minimum value of 2sinx+2cosx over real x.
Solution: By AM-GM, 2sinx+2cosx≥22sinx+cosx. The exponent sinx+cosx has minimum −2, so the right side is at least 22−2=21−1/2. Both inequalities become equalities at the same point, x=45π, where sinx=cosx=−21. So the minimum is 21−1/2. When AM-GM is chained with another bound, the answer is valid only if both equality conditions can hold together, which they do here.
Brain-Teaser 2: If a,b,c>0 and a+b+c=1, find the minimum of a1+b1+c1 and the maximum of ab+bc+ca.
Solution: By AM-HM, (a+b+c)(a1+b1+c1)≥9, so the sum of reciprocals is at least 9, attained at a=b=c=31. For the second, (a+b+c)2=a2+b2+c2+2(ab+bc+ca) and a2+b2+c2≥ab+bc+ca (which is 21∑(a−b)2≥0). So 1≥3(ab+bc+ca), giving a maximum of 31 at a=b=c=31. Symmetric problems usually have their extremes at the symmetric point, but the inequality proves it.
Brain-Teaser 3: A rectangular box without a lid has volume 32 cubic units. Find the dimensions that minimise the surface area.
Solution: With base x×y and height h, xyh=32 and the area is S=xy+2xh+2yh. Apply AM-GM to the three terms: S≥33xy⋅2xh⋅2yh=334(xyh)2=334×1024=3×16=48. Equality needs xy=2xh=2yh, so x=y=2h; with xyh=4h3=32, h=2 and x=y=4. The minimum area is 48. Choosing the three area terms as the quantities to compare is what makes their product a function of the fixed volume only.
