AM-GM Inequality

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Explanations 14

The Arithmetic Mean-Geometric Mean inequality is an inequality that states a1+a2+a3⋯ann≥a1a2a3⋯ann\dfrac{a_1+a_2+a_3\cdots a_n}{n}\geq\sqrt[n]{a_1a_2a_3\cdots a_n}na1+a2+a3⋯a…
By Hyunwoo (David) Lee, Someone Someone, Jason Li
The AM GM Inequality is that the Arithmetic Mean is always greater than or equal to the Geometric Mean. AM≥GMAM\ge GMAM≥GM
By Ryan Jiang
AM-GM The arithmetic mean, or average as we call it, of the numbers a1,a2,…,ana_1,a_2,\dots,a_na1,a2,…,an is equal to AM=a1+a2+⋯+annAM=\frac{a_1+a_2+\dots+a_n}{n}AM=na1+a2+…
By Cody Johnson
Help, I Broke AM-GM! What is wrong with the following applications of AM-GM? x−2−1x3≥x⋅(−2)⋅(−1x)3=23\frac{x-2-\frac1x}3\ge\sqrt[3]{x\cdot(-2)\cdot\left(-\frac1x\right)}=\sqrt[3…
By Cody Johnson, Yong See Foo
Inequalities (Part 2) AM-GM inequality This is a very useful inequality. Most problems requires an application of this inequality. The inequality states that : For all positive …
By Zi Song Yeoh, Alex Federspiel
The AM-GM inequality for two variables states that for two positive real numbers aaa and bbb, a+b2≥ab.\dfrac{a+b}{2} \geq \sqrt{ab}.2a+b≥ab. The left side is called the arithm…
By Vinjai Vale, Yan Yau Cheng, Joshua Siktar
Solving inequalities is a very useful skill in algebra. Let us start with the trivial inequality. The trivial inequality states that x2≥0x^2\geq0x2≥0 for real xxx. This seemingl…
By Jacob Nie, Catherine Yeo, Ed Guzovsky
Prove that if a1a2⋯an=1a_1a_2\cdots a_n=1a1a2⋯an=1 and the aka_kak are non-negative, then a1+a2+⋯+an≥n.a_1+a_2+\cdots+a_n\geq n.a1+a2+⋯+an≥n.
By Jacob Nie, Ed Guzovsky, Moor Xu
Prove that, for a,b,c>0a, b, c > 0a,b,c>0, (abc+a)(b+c)≥4abc(abc + a)(b + c) \geq 4abc(abc+a)(b+c)≥4abc
By Joshua Siktar
Prove that: 4a+9b≥25a+b\frac{4}{a}+\frac{9}{b}\geq \frac{25}{a+b}a4+b9≥a+b25 By Cauchy Schwartz: (a+b)(4a+9b)≥(3+2)2(a+b)\left(\frac{4}{a}+\frac{9}{b}\right)\geq (3+2)^2(a+b)…
By Yan Yau Cheng
Prove that, for x>0x > 0x>0 and m,n>0m, n > 0m,n>0, that xn+xm≥2xm+n2x^{n} + x^{m} \geq 2x^{\frac{m + n}{2}}xn+xm≥2x2m+n
By Joshua Siktar
Prove that for any positive real numbers a,b,ca, b, ca,b,c, a4+4b4+c2≥4abc.a^4 + 4b^4 + c^2 \geq 4abc.a4+4b4+c2≥4abc. Applying the AM-GM inequality, we see that a4+4b4≥2a4⋅4b4=4…
By Vinjai Vale
So AM GM inequality is used to prove many stuffs especially problems in Olympiads it is the basics of inequalities.
By Aditya Thomas
So, to recap: There's n positive numbers whose product is one, and we wanna prove their sum is at least n. Simple! Their geometric mean is 1, since the nth root of one is one. B…
By Ishwar Karthik

Problems 8