This GeoGebra activity shows four different [i]means[/i] of two real, non-negative values [color=#0000ff][i]a[/i][/color], [color=#ff0000][i]b[/i][/color]:[br][color=#b20ea8][i]- AM [/i]= arithmetic[/color], [color=#198f88][i][br]- GM[/i] = geometric[/color], [br]- [color=#0a971e][i]QM[/i] = quadratic[/color][color=#d69210],[i][br]- HM[/i] =harmonic[/color] [br]geometrically.[br][br]The values are dynamically shown as segments on the diameter of a semicircle, and can be modified by moving point X.[br][br]Move point [i]X[/i] to explore the means of [i]a[/i] and [i]b[/i], and explore the inequality [br][math]QM\left(a,b\right)\ge AM\left(a,b\right)\ge GM\left(a,b\right)\ge HM\left(a,b\right)[/math].

Recalling the expressions that define the means of two numbers [math]a,b\in\mathbb{R}^+_0[/math]:[br]- quadratic [math]QM=\sqrt{\frac{a^2+b^2}{2}}[/math][br]- arithmetic [math]AM=\frac{a+b}{2}[/math][br]- geometric [math]GM=\sqrt{ab}[/math][br]- harmonic [math]HM=\frac{2ab}{a+b}[/math][br]we can conclude that the following inequality holds:[br] [math]\sqrt{\frac{a^2+b^2}{2}}\ge\frac{a+b}{2}\ge\sqrt{ab}\ge\frac{2ab}{a+b}[/math].[br]

The AM-GM inequality is particularly known in its general form: given [i]n[/i] real numbers [math]x_1,x_2,...,x_n\ge0[/math], we have that [math]\frac{x_1+x_2+...+x_n}{n}\ge\sqrt[n]{x_1\cdot x_2\cdot...\cdot x_n}[/math].[br]