Answer to Question #15640 in Abstract Algebra for Aziza

Question

Suppose 2a + 5b = 10 for some nonnegative real numbers a and b. The geometric mean of a and b attains its maximum when b is equal to which number?

Answer 1

Let's write down the formula for geometric mean of a and b:

G = √(ab).

We know that

2a + 5b = 10 ==> a = 5 - 2.5b,

so

G = √(ab) = √((5 - 2.5b)b) = √(5b - 2.5b²).

G reachs its maximum value when 5b - 2.5b² do, so let's maximise it with respect to b:

(5b - 2.5b²)' = 5 - 5b;

5 - 5b = 0 <==> b = 1.

Therefore, b=1 maximises G.