1) Find two real numbers whose sum is 30 and whose product is maximized.

2) Find two numbers whose difference is 50 and whose product is minimized.

1) Both 15.

2) 25 and - 25.

Step-by-step explanation:

1) Let the 2 numbers be x and 30 - x.

The product = x (30 - x)

f (x) = x (30 - x)

f (x) = 30x - x^2

Finding the derivative:

f' (x) = 30 - 2x

Finding the maximum:

30 - 2x = 0

x = 15.

This gives a maximum f (x) because f" (x) = - 2.

So the numbers are 15 and 30 - 15 = 15.

2). If one number is x the other is y.

x - y = 50

y = x - 50

The product =

x (x - 50)

= x^2 - 50x

Finding the derivative:

2x - 50 = 0 for a minimum value.

2x = 50

x = 25.

So the numbers are 25 and 25-50 = - 25.