Question

given y = 4x-3

find the minimum value of xy.
find the rate of change of xy with respect to x.
find the rate of change of xy with respect to y.

Answer 1

f(x,y) = xy = x (4x-3)
f = 4 x^2 - 3x
df/dx = 8 x - 3
that is 0 when x = 3/8 and the second derivative is positive so it is a minumum
then x y = (3/8)(3/2 -3)
= (3/8) (-3/2) = -9/16

we found df/dx

now df/dy
4x = y+3
x = (y+3)/4
g(y) = xy = y(y+3)/4
= (1/4)(y^2+3y)
dg/dy = (1/4)(2 y+3)
= y/2 + 3/4