Question

Suppose 2a + 5b = 10 for some nonnegative real numbers a and b. What is the largest number that the arithmetic mean of a and b can be? (You may want to use some of the techniques from Calc I.)

Answer 1

Let

m = (a+b)/2

be the arithmetic mean of a and b.

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

m = (a+b)/2 = (5 - 2.5b + b)/2 = (5 - 1.5b)/2 = 2.5 - 0.75b.

m reaches its maximum value when b is equal to zero. So, the largest number that the arithmetic mean can be is 2.5.