ALG-03

AM–HM inequality

(x+y)/2 ≥ 2xy/(x+y) for x,y > 0. AM ≥ HM.

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InequalitiesAM–HM

Governing equation

\dfrac{x+y}{2}\ge\dfrac{2xy}{x+y}

$x \in \left[0.1, 40\right]$

where

x: x (—)
y: y (—)
AM: Arithmetic mean (—)
HM: Harmonic mean (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\dfrac{x+y}{2}\ge\dfrac{2xy}{x+y}

$x \in \left[0.1, 40\right]$

$x = 0.1$ $AM = 0.198$

Along the curve

②

Domain of the variable — blue

x_{\min} = 0.1,\quad x_{\max} = 40

$x \in \left[0.1, 40\right]$

$x = 0.1$ $x = 0.1$

Along the curve

③

Response curve — green

\dfrac{x+y}{2}\ge\dfrac{2xy}{x+y}

$x \in \left[0.1, 40\right]$

$x = 0.1$ $AM = 0.198$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}AM}{\mathrm{d}x}

$x \in \left[0.1, 40\right]$

$x = 0.351$ $AM^{\prime} = 1.889$

Along the curve

Lecture brief

Historical brief

From al-Khwārizmī’s restoration to the quadratic formula, AM–GM, De Moivre and logarithms, algebra is the closed-form skeleton under every lab. These sheets solve, bound and compound. This sheet (ALG-03 — AM–HM inequality) is the form associated with AM–HM. Working symbols: $x$ , $y$ $\rightarrow$ $AM$ , $HM$ . The harmonic mean is the reciprocal of the mean of reciprocals. Equality iff x = y.

Purpose

Purpose: compute

AM

HM

from

x

y

in Algebra via

\dfrac{x+y}{2}\ge\dfrac{2xy}{x+y}

(x+y)/2 ≥ 2xy/(x+y) for x,y > 0. AM ≥ HM. Use it when a real algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

x = 4.000\,\mathrm{—}

y = 12.000\,\mathrm{—}

, the governing relation

\dfrac{x+y}{2}\ge\dfrac{2xy}{x+y}

yields

AM = 8.000\,\mathrm{—}

HM = 6.000\,\mathrm{—}

. Two positive numbers. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

x x4 —y y12 —

Outputs

Arithmetic mean AM8.000 —
Harmonic mean HM6.000 —

Reading speed

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Narration of this film

Two positive numbers.

The harmonic mean is the reciprocal of the mean of reciprocals. Equality iff x = y.

Reading speed

Watch on YouTube