minimising???

Question

i need to minimise eqn 1 subject to eqn 2
(x +12)(y + 8) ---> eqn 1
x*y = 384 ------> eqn 2

Answer 1

xy = 384 ==> y = 384/x

(x + 12)(y + 8) = (x + 12)((384/x) + 8) = 8x + 384 + 96 + 12*384/x = 8x + 480 + 4608/x

To minimize an expression like this, find all points where the derivative is equal to zero; this identifies all of the local extremes. Then check the points and see which is the minimum.

d(8x + 480 + 4608/x)/dx = 8 - 4608/x^2 = 0 ==> 4608/x^2 = 8 ==> x^2 = 4608/8 ==> x = sqrt(576) = 24. x = 24 is the only point where the derivative is zero. x = 24 ==> y = 384/24 = 16. Let's check the value of equation 1. (x + 12)(y + 8) = (24 + 12)(16 + 8) = 36*24 = 864. Considering that xy = 384, and each of our new factors are larger than x and y, it makes sense that the minimum must be larger than 384.

Answer 2

(x +12)(y + 8) ---> xy + 8x + 12y + 96
x*y = 384 ------> implies that you need to minimise,

480 + 8x + 12y

if there are no restrictions on x,y then this can take any value in (-oo,oo)