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Ex 4.3

3, Find the maximum product of two numbers (not necessarily integers) that have a sum of 100.

Solutions: The problem states that we are looking for two numbers

The problem states that we are looking for two numbers x and y such as x + y = 100

that is y = 100 - x

We would like to find where the product xy is maximum, but from the above equation we can write:

xy = x(100 - x) = 100x - x^2

So we now have a one-variable function

f(x) = 100x - x^2, and must find a positive value of x where the function f reaches a maximum.

To do that we calculate the derivative f’(x) = 100 - 2x and we look for values of x where f’(x) = 100 - 2x = 0

x = 50

The solution is then x = 50, y = 50