Given two positive numbers a and b, we define the geometric mean (G.M.) and the arithmet...

Given two positive numbers a and b, we define the geometric mean (G.M.) and the arithmetic mean (A.M.) as follows:

(a) Complete the table, using a calculator as necessary so that the entries in the third and fourth columns are in decimal form.

(b) Prove that for all nonnegative numbers a and b we have

Hint: Use the following property of inequalities:

If x and y are nonnegative, then the inequality is equivalent to

(c) Assuming that a = b (and that a and b are nonnegative), show that inequality (1) becomes an equality.

(d) Assuming that a and b are nonnegative and that

Remark: Parts (b) through (d) can be summarized as follows. For all nonnegative numbers a and b, we have

with equality holding if and only if a = b.

This result is known as the arithmetic-geometric mean inequality for two numbers. The mini project at the end of this section shows an application of this result.