Optimization Ap Calc Problem?

Question

Find 2 positive numbers that satisfy the given requirements
[Two separate problems ]

1.) The sum S and the product is a maximum

2.) The sum of the first and twice the second is 100 and the product is the maximum

Thanks.

Answer 1

1)

a+ b = S
(where a and b are two positive integers)

a = S - b

product, P = a* b = (S-b)(b) = sb - b^2

P will be maximum , when dP/db = 0

differentiate P with respect to b

dP/db = S - 2b

when dP/db = 0

S = 2b

b = S/2

a = S/2

2)

let a = first

b = second

a + 2b = 100

a = 100 -2b

Now product, P = ab

P = (100- 2b)(b)

P = 100b - 2b^2

P will be maximum when dP/db = 0

dP/db = 100 - 4b

100 - 4b = 0

4b = 100

b = 100/4 = 25

a = 100 - (2*25) = 50

the first is 50 and second is 25

Answer 2

2) x+2y=100 z=xy max

x= 100-2y so z= (100-2y)*y = 100y -2y^2
z´= 100-4y=0 so y = 25 and x=50
1) The question is not clear