Question

Minimize S = x + 19y with xy = 19 and both x and y > 0.
s=?

Minimize F = 8x2 + 8y2 with xy2 = 16.
F=?

Maximize P = xyz with x + y = 3 and y + z = 3, and x, y, and z greater than or equal to 0.
P=?

Answer 1

these are optimization problems, to minimize or maximize a function, find the point where derivative equals zero
for each problem , get function in terms of a single variable, then differentiate and set equal to 0. then solve for value of variable. Use this to get optimized function value.

here is how the last one is solved:
P = xyz
y = 3-x
z = 3-y = 3-(3-x) = x
therefore,
P = x*(3-x)*x = 3x^2 - x^3
differentiate
dP/dx = 6x-3x^2
set equal to 0
--> 6x - 3x^2 = 0
--> 3x(2-x) = 0
--> x = 0, x=2
if x=0, then P=0....but that is not a maximum, so use x=2
if x=2, then P = 4

Maximum value of P is 4