- #1
Inertigratus
- 128
- 0
Homework Statement
Find the maximum and minimum value of the function, defined over x2 + y2 + z2 [itex]\leq[/itex] 1.
x [itex]\geq[/itex] 0, y [itex]\geq[/itex] 0, y [itex]\geq[/itex] 0.
Homework Equations
f(x,y,z) = xy(z+1)
The Attempt at a Solution
[itex]\nabla[/itex]f = (y(z+1), x(z+1), xy) = 0
Gets me (0, y, -1), (x, 0, -1), (0, 0, z) and they all result in f(x,y,z) = 0.
Then I wasn't sure how to find the values on the sphere.
What I did was I switched to spherical coordinates with r = 1 and plugged them into the eq.
f([itex]\theta[/itex], [itex]\varphi[/itex]) = sin2[itex]\theta[/itex](cos[itex]\theta[/itex] + 1)cos[itex]\varphi[/itex]sin[itex]\varphi[/itex].
Then it's rather obvious that to get max, [itex]\theta[/itex] = +-[itex]\pi[/itex]/2 and [itex]\varphi[/itex] = [itex]\pi[/itex]/4.
Plugging that back into the cartesian coordinates and into the function gives +- 1/2.
Maximum is supposed to be 16/27 and minimum 0.
By the way, this problem comes before the problems that are about optimizing functions with constraints. So no need to use the lagrange multiplier.
Any ideas? :)