Let a, b, and c be nonnegative numbers. Follow steps (a) through (e) to show thatwith equa...

Let a, b, and c be nonnegative numbers. Follow steps (a) through (e) to show that

with equality holding if and only if a = b = c.

This result is known as the arithmetic-geometric mean inequality for three numbers. (Applications are developed in the projects at the ends of Section.)

(a) By multiplying out the right-hand side, show that the following equation holds for all real numbers A, B, and C.

(b) Now assume for the remainder of this exercise that A, B, and C are nonnegative numbers. Use equation (1) to explain why

(c) Make the following substitutions in inequality (2): A3 = a, B3 = b, and C3 = c. Show that the result can be written

(d) Assuming that a = b = c, show that inequality (3) becomes an equality.

(e) Assuming , show that a = b = c.

Hint: In terms of A, B, and C, the assumption becomes . Use this to substitute for ABC on the left-hand side of equation (1). Then use the resulting equation to deduce that A = B = C, and consequently a = b = c.