Question

Find two positive numbers A and B (with A≤B) whose sum is 44 and whose product is maximized

Answer 1

so A+B=44
now let P be the product of the two so
P=AB
notice that using the first equation, B=44-A, so the product becomes
P=A(44-A)
P=44A-A^2
now this is just a single variable calc. problem about optimization. So we find the critical points(set P'=0 and find the A there).
P'=44-2A
P'=0
44-2A=0
-2A=-44
A=22
since P''(22)<0 then we know A=22 is a maximum..
so the first number is x=22 and the second number will be
B=44-A=44-22=22
So A=22 and B=22 will yield the maximum product

Answer 2

sorry typographical error.. change the x at the line " so the first number is x=22 and the second number will be B=44-A=44-22=22" to A

Answer 3

got used to x :))

Answer 4

Find two numbers A and
B (with
A≤B) whose difference is 38 and whose product is minimized

if it had asked me to do minimized

Answer 5

ok notice that this will be a single var. calc optimization prob again. Here we have
B-A=38
let P be the product of the two
P=AB
we know that
P=A(A+38)
P=A^2+38A
find the critical points again:
P'=2A+38
P'=0
2A+38=0
2A=-38
A=-19
this will be the minimum since P''(-19)>0...
so B will be:
B=38+A=38-19=19
the answer will be B=19 and A=-19.