SOLUTION: Consider the ellipse x^2 + 2y^2 = 16. Find the maximum value of the product xy on the ellipse.

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Consider the ellipse x^2 + 2y^2 = 16. Find the maximum value of the product xy on the ellipse.      Log On


   

Question 883874: Consider the ellipse x^2 + 2y^2 = 16. Find the maximum value of the product xy on the ellipse.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2%2B2y%5E2=16
x%5E2=16-2y%5E2
x=sqrt%2816-2y%5E2%29
Substitute,
z=xy
z=sqrt%2816-2y%5E2%29%2Ay
Take the derivative of z with respect to y and set it equal to zero.
dz%2Fdy=sqrt%2816-2y%5E2%29%2B%28y%28-2y%29%29%2Fsqrt%2816-2y%5E2%29
dz%2Fdy=%2816-y%5E2%29%2Fsqrt%2816-2y%5E2%29%2B%28-2y%5E2%29%2Fsqrt%2816-2y%5E2%29
dz%2Fdy=%28%2816-2y%5E2-2y%5E2%29%29%2Fsqrt%2816-2y%5E2%29
dz%2Fdy=%2816-4y%5E2%29%2Fsqrt%2816-2y%5E2%29
%2816-4y%5E2%29%2Fsqrt%2816-2y%5E2%29=0
4y%5E2=16
y%5E2=4
y=2 and y=-2
Then,
x=sqrt%2816-8%29=sqrt%288%29
So,
z%5Bmax%5D=2sqrt%288%29