Classical Inequalities

Concept Quizzes

Power Mean Inequalities
Applying the Arithmetic Mean Geometric Mean Inequality

Challenge Quizzes

Classical Inequality Statements: Level 2 Challenges
Classical Inequalities: Level 4 Challenges

Power Mean Inequalities

The arithmetic mean $A = \frac{a+b}{2}$ and the geometric mean $G = \sqrt{ab}$ of non-negative quantities $a$ and $b$ are shown. Based on the diagram, which inequality is correct?

\frac{a+b}{2} \geq \sqrt{ab}

\sqrt{ab} \geq \frac{a+b}{2}

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If the sum of two positive numbers is 18, what is the largest possible value of their product?

Hint. For non-negative numbers $a$ and $b,$ the Arithmetic Mean - Geometric Mean Inequality implies that $\frac{a+b}{2} \geq \sqrt{ab},$ and $\frac{a+b}{2} = \sqrt{ab} \text{ when } a = b.$

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by Brilliant Staff

For values of $x$ such that $(8 +x)$ and $(1 -x)$ are positive, what is the maximum possible value of

$(8 +x)(1-x)?$

Hint. Apply the Arithmetic Mean - Geometric Mean Inequality.

4.5

8

10

20

\frac{81}{4}

\frac{41}{2}

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by Brilliant Staff

If the sum of two legs of a right triangle is 20, what is the smallest possible value of the hypotenuse?

Hint. The quadratic mean, or root-mean-square of non-negative numbers $a$ and $b$ is greater than or equal to their arithmetic mean, i.e.

$\sqrt{\frac{a^2+b^2}{2}} \geq \frac{a+b}{2},$

with equality happening when $a = b.$

5\sqrt{2}

10

10\sqrt{2}

10\sqrt{3}

15

15.38

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by Brilliant Staff

What is the quadratic mean of the four values $0, 0, 0,$ and $8?$

Note. The quadratic mean, or root-mean-square, of non-negative numbers $x_1, \ldots, x_n$ is

$\sqrt{\frac{x_1^2+\cdots + x_n^2}{n}}.$

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by Brilliant Staff