Means | Articles containing proofs | Inequalities
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which case they are both that number). The simplest non-trivial case – i.e., with more than one variable – for two non-negative numbers x and y, is the statement that with equality if and only if x = y. This case can be seen from the fact that the square of a real number is always non-negative (greater than or equal to zero) and from the elementary case (a ± b)2 = a2 ± 2ab + b2 of the binomial formula: Hence (x + y)2 ≥ 4xy, with equality precisely when (x − y)2 = 0, i.e. x = y. The AM–GM inequality then follows from taking the positive square root of both sides and then dividing both sides by 2. For a geometrical interpretation, consider a rectangle with sides of length x and y, hence it has perimeter 2x + 2y and area xy. Similarly, a square with all sides of length √xy has the perimeter 4√xy and the same area as the rectangle. The simplest non-trivial case of the AM–GM inequality implies for the perimeters that 2x + 2y ≥ 4√xy and that only the square has the smallest perimeter amongst all rectangles of equal area. Extensions of the AM–GM inequality are available to include or generalized means. (Wikipedia).
Arithmetic geometric mean inequality
In this video I give an elementary proof of the arithmetic-geometric mean inequality using calculus 1 techniques. This inequality states that the arithmetic mean of a list of positive numbers is always greater than or equal to its geometric mean. I would like to thank Alex Zorba for provi
From playlist Calculus
Learn about the geometric mean of numbers. The geometric mean of n numbers is the nth root of the product of the numbers. To find the geometric mean of n numbers, we first multiply the numbers and then take the nth root of the product.
From playlist Geometry - GEOMETRIC MEAN
How to determine the geometric mean between two numbers
Learn about the geometric mean of numbers. The geometric mean of n numbers is the nth root of the product of the numbers. To find the geometric mean of n numbers, we first multiply the numbers and then take the nth root of the product.
From playlist Geometry - GEOMETRIC MEAN
Solving and Graphing an inequality when the solution point is a decimal
👉 Learn how to solve multi-step linear inequalities having parenthesis. An inequality is a statement in which one value is not equal to the other value. An inequality is linear when the highest exponent in its variable(s) is 1. (i.e. there is no exponent in its variable(s)). A multi-step l
From playlist Solve and Graph Inequalities | Multi-Step With Parenthesis
Ex: Determine if a Sequence is Arithmetic or Geometric (arithmetic)
This video provides two examples of how to determine if a sequence is arithmetic or geometric. These two examples are arithmetic. Site: http://mathispower4u.com Blog: http://mathispower4u.wordpress.com
From playlist Sequences
What is the formula to find the sum of an arithmetic sequence
👉 Learn all about series. A series is the sum of the terms of a sequence. Just like in sequences, there are many types of series, among which are: arithmetic and geometric series. An arithmetic series is the sum of the terms of an arithmetic sequence. A geometric series is the sum of the t
From playlist Series
Problem-Solving Trick No One Taught You: RMS-AM-GM-HM Inequality
This inequality is famous in math competitions and in theoretical proofs. But why is it true? The video presents a great geometric visualization and proof for two variables. Pay attention--I'll use this inequality in an upcoming video! Desmos.com link https://www.desmos.com/calculator/6kb
From playlist Mathematical Proofs
OVERKILL | an obvious inequality
We prove an "obvious" compound inequality of four numbers by first establishing the harmonic-geometric-arithmetic-quadratic mean inequality. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.com/stores/michael-penn-math Personal Website
From playlist Overkill | Solving simple problems with overpowered math.
Learn how to solve a multi step inequality and graph the solution
👉 Learn how to solve multi-step linear inequalities having parenthesis. An inequality is a statement in which one value is not equal to the other value. An inequality is linear when the highest exponent in its variable(s) is 1. (i.e. there is no exponent in its variable(s)). A multi-step l
From playlist Solve and Graph Inequalities | Multi-Step With Parenthesis
The arithmetic-geometric mean inequality
I’m back!!! Thanks for your patience, I finally found the time to upload more YouTube videos, yaaaay! In this video I present a proof of the Arithmetic-Geometric mean inequality using Lagrange multipliers. Enjoy!
From playlist Partial Derivatives
This video aims to enlighten students interested in competition maths/olympiads on this very useful inequality. The video covers a couple of different ways to prove the inequality along with a simple example.
From playlist Summer of Math Exposition Youtube Videos
Mean Inequalities via Moments #SoME1 (visual proof)
This short animated proof demonstrates the two variable mean inequalities by use of moments of mass. #mathshorts #mathvideo #math #inequality #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #mathematics #physics #moments #weight #3b1bsom
From playlist Inequalities
We present a solution to question B2 from the 2005 William Lowell Putnam Mathematics Competition. Our solution uses a lesser known inequality known as the harmonic mean - arithmetic mean inequality (HM-AM). http://www.michael-penn.net https://www.researchgate.net/profile/Michael_Penn5 ht
From playlist Putnam Exam Solutions
A Central American inequality.
We look at a nice problem from the Centro-American mathematics olympiad. Suggest a problem: https://forms.gle/ea7Pw7HcKePGB4my5 Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.com/stores/michael-penn-math Personal Website: http://www
From playlist Math Contest Problems
Learn to solve and graph a inequality with a fraction learn two different ways
👉 Learn how to solve multi-step linear inequalities having parenthesis. An inequality is a statement in which one value is not equal to the other value. An inequality is linear when the highest exponent in its variable(s) is 1. (i.e. there is no exponent in its variable(s)). A multi-step l
From playlist Solve and Graph Inequalities | Multi-Step With Parenthesis
A. Chambert-Loir - Equidistribution theorems in Arakelov geometry and Bogomolov conjecture (part3)
Let X be an algebraic curve of genus g⩾2 embedded in its Jacobian variety J. The Manin-Mumford conjecture (proved by Raynaud) asserts that X contains only finitely many points of finite order. When X is defined over a number field, Bogomolov conjectured a refinement of this statement, name
From playlist Ecole d'été 2017 - Géométrie d'Arakelov et applications diophantiennes