Question

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.

Answer 1

Answer:

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.