ALG-02

AM–GM inequality

(x+y)/2 ≥ √(xy) for x,y ≥ 0, equality iff x = y.

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InequalitiesAM–GM

Governing equation

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

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

where

x: x (—)
y: y (—)
AM: Arithmetic mean (—)
GM: Geometric 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\sqrt{xy}

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

$x = 0.1$ $AM = 0.949$

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\sqrt{xy}

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

$x = 0.1$ $AM = 0.949$

Along the curve

④

Rate of change — amber

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

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

$x = 0.351$ $AM^{\prime} = 2.747$

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-02 — AM–GM inequality) is the form associated with AM–GM. Working symbols: $x$ , $y$ $\rightarrow$ $AM$ , $GM$ . The arithmetic mean is at least the geometric mean — Jensen on ln, or (√x−√y)² ≥ 0.

Purpose

Purpose: compute

AM

GM

from

x

y

in Algebra via

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

(x+y)/2 ≥ √(xy) for x,y ≥ 0, equality iff x = y. 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 = 9.000\,\mathrm{—}

, the governing relation

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

yields

AM = 6.500\,\mathrm{—}

GM = 6.000\,\mathrm{—}

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

Calculator

Inputs

x x4 —y y9 —

Outputs

Arithmetic mean AM6.500 —
Geometric mean GM6.000 —

Reading speed

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Narration of this film

Two positive numbers, two means.

The arithmetic mean is at least the geometric mean — Jensen on ln, or (√x−√y)² ≥ 0.

Reading speed

Watch on YouTube