Waste less time on Facebook — follow Brilliant.
×
Algebra

Classical Inequalities

Concept Quizzes

Challenge Quizzes

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?

View solution
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 \[\frac{a+b}{2} = \sqrt{ab} \text{ when } a = b.\]

View solution
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.

View solution
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 \(a = b.\)

View solution
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

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

View solution
View wiki
by Brilliant Staff
×

Problem Loading...

Note Loading...

Set Loading...