How to calculate the minimum and maximum values in a range?

Question

I have the next function

$f(x, y) = x^4 + y^4-8 (x^2 + y^2)$ at $x^2 + y^2 = 1$

What are the steps to calculate the maximum and minimum points of the function in that range?

Answer 1

Try polar coordinates. Let $x = \cos\phi, y = \sin\phi$. Then you are finding the minimum/maximum for $$ f(\phi) = (\cos\phi)^4 + (\sin\phi)^4 - 8 $$ when $\phi \in ]-\pi,\pi]$. Using power-reduction formulae and some algebra, $$ f'(\phi) = -\sin 4\phi $$ You can use Fermat's theorem to finish the proof.

Answer 2

If you are only looking for extremal values on the circle (and not within the disk), you can replace the last term at once by 8 . The "constraint equation" gives $ \ y^2 \ = \ 1 \ - \ x^2 \ $ , so you are left to find the extremal values of a single-variable function $ \ x^4 \ + \ (1 \ - \ x^2)^2 \ - \ 8 \ $ on $ \ [ -1, \ 1 ] \ $. (Don't neglect to check the endpoints!]

Note that both the function and the circle are symmetric about the origin, so there are pairs of points with the extremal values.

Answer 3

First, $(x^2+y^2)^2 = x^4 + y^4 +2x^2y^2.$ So for $x^2+y^2=1$ you have $f(x,y) = 1 - 2x^2y^2 - 8 = -7-2x^2(1-x^2)$ where $x^2$ can take any value from $-1$ to $1$. Substitute $z = x^2$; $z$ ranges from $0$ to $1$. Hope you can complete the exercise.