Comparison of means
This GeoGebra activity shows four different means of two real, non-negative values a, b:
- AM = arithmetic,
- GM = geometric,
- QM = quadratic,
- HM =harmonic
geometrically.
The values are dynamically shown as segments on the diameter of a semicircle, and can be modified by moving point X.
Move point X to explore the means of a and b, and explore the inequality
.
Recalling the expressions that define the means of two numbers :
- quadratic
- arithmetic
- geometric
- harmonic
we can conclude that the following inequality holds:
.
A question:
For which values of a and b the inequality of means is an equality?
The AM-GM Inequality
The AM-GM inequality is particularly known in its general form: given n real numbers , we have that .