Excel in math and science.

Join using Facebook Join using Google Join using email

Forgot password? New user? Sign up

Existing user? Log in

Your answer seems reasonable. Find out if you're right!

That seems reasonable. Find out if you're right!

Join using Facebook Join using Google Join using email

Forgot password? New user? Sign up

Existing user? Log in

Algebra

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}

Show explanation

View wiki

by Brilliant Staff

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 $\frac{a+b}{2} = \sqrt{ab} \text{ when } a = b.$

Show explanation

View wiki

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}

Show explanation

View wiki

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

Show explanation

View wiki

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

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

Show explanation

View wiki

by Brilliant Staff