SOLUTION: Consider all the points in the plane that solve the equation x^2 + 2y^2 = 16. Find the maximum value of the product xy on this graph.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons -> SOLUTION: Consider all the points in the plane that solve the equation x^2 + 2y^2 = 16. Find the maximum value of the product xy on this graph.      Log On


   

Question 1027081: Consider all the points in the plane that solve the equation x^2 + 2y^2 = 16. Find the maximum value of the product xy on this graph.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Z=xy
To simplify, the product of Z%5E2 would also be a maximum so find the maximum value of Z%5E2=%28x%5E2y%5E2%29.
From the equation,
x%5E2=16-2y%5E2%29
Substituting,
Z%5E2=%2816-2y%5E2%29y%5E2
g%28y%29=Z%5E2=16y%5E2-2y%5E4
Find the derivative of g=Z%5E2,
dg%2Fdy=32y-8y%5E3
32y-8y%5E3=0
8y%284-y%5E2%29=0
Two possible solutions:
y=0
This is not a maximum but a local minimum (check with second derivative test).
4-y%5E2=0
y%5E2=4
y=0+%2B-+2
Then,
x%5E2=16-2%282%29%5E2
x%5E2=8
x=0+%2B-+2sqrt%282%29
So the maximum value would be,
xy=2%2A2%28sqrt%282%29%29=4sqrt%282%29