A.) Show that z=3x+1200/x. B.) Find the stationary value of z and determine if it's a maximum or minimum.

Question

The variable x,y, and z can take the only positive values and are such that z=3x+2y, and xy=600

Answer 1

z=3x+2y and xy=600. Note that xy=600 can be re-arranged to be y=600/x and that can be substituted into z=3x+2y to result in z=3x+1200/x ... dz/dx = 3-1200/x². Setting that =0 and solving for x yields x=±20. Using only the + value yields a stationary point at x=20. z''=2400/x³ and is >= at x=20, thus the stationary point is a minimum.