You must be logged in to see worked solutions.

Already have an account? Log in here.

Your destination for questions of the form "do we know if this expression is always greater than this other expression?" Explore what humans know about mathematical 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?

You must be logged in to see worked solutions.

Already have an account? Log in here.

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

You must be logged in to see worked solutions.

Already have an account? Log in here.

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.

You must be logged in to see worked solutions.

Already have an account? Log in here.

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 \(a = b.\)

You must be logged in to see worked solutions.

Already have an account? Log in here.

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

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

You must be logged in to see worked solutions.

Already have an account? Log in here.

×

Problem Loading...

Note Loading...

Set Loading...