I'm trying to solve a math problem and have reduced it to finding the maximum and minimum values of the function specified in the title of this post, and I have no idea how I can derive simple analytical expressions for those values.
It's a homework problem for university students, so the answer and the solution are supposed to be simple.
By introducing $x = \cos \phi$, I can obtain a fourth-order polynomial equation for the positions of the maximum and minimum: $$ x^4-6x^3+24x^2+6x-9=0. $$ But I don't see how I can analytically find its roots. I know that each fourth-order polynomial equation can be solved analytically by using some extremely complicated formulas, but students are surely not supposed to bother using them.
Does it look like the lecturer made a typo in the formulation of the problem, or is there a simple solution?
P.S. Just in case, here's the original problem:
Find the maximum and minimum values of $w(x,y,z) = 6x - x^2 - 8 y + z^3$ under the condition $x^2 + y^2 + z^2 \leq 1$.
I proved and also numerically verified that that those values are reached somewhere at the equator of the sphere, that is, at $z=0$, $x^2 + y^2 = 1$. This is how the problem is reduced to finding the maximum and minimum values of the function specified in the title of this post.