Optimization Calculus Question

Question

Find two numbers A and B (with A<= B) whose difference is 42 and whose product is minimaized.

A = ???

B = ???

Answer 1

A-B = 42

minimize AB

B= A -42

minimize A(A-42) = A^2 -42A

take the derivative and set equal to zero,

2A -42 = 0

A = 42/2 = 21

B = 21-42 = -21

A = 21, B = -21

-441 is the minimum AB

A^2 -42A is an upwarding opening parabola with vertex (21 ,-441) = minimum point with -441 the minimum

one number = 21, the other -21 to get a minimum product of -441