maximum of a function?
At first, you have to find a math book about function with more than one variable. ANALYSES MATH BOOK.
Here is just a small introduction about this topic. You know, this is a chapter in my math book.
Let's take an example for you to take the overview: remember that we can not solve the others like this. They have cases.
We have a function with 2 variables: f(x,y)=2x^4 + y^4 - x^2 - 2y^2
Then:
df/dx = 8x^3 - 2x = 0
df/dy = 4y^3 - 4y = 0
we have 9 points to determine
P1(0,0)
P2(0,1)
P3(0,-1)
P4(1/2,0)
P5(-1/2,0)
P6(1/2,1)
p7(1/2,-1)
P8(-1/2,1)
P9(-1/2,-1)
Besides
d^2f/dx^2 = 24x^2 - 2 = A
d^2f/dxdy = 0 = B
d^2f/dy^2 = 12^2 - 4 = C
So at P1, f is maximum with close meaning (just maximum to all the points near P1)
If you want to know the maximum of f beyond RxRxR, you'll have to do more.
at P6,P7,P8,P9 f is minimum with close meaning
at P2, P3, P4, P5 we don't have any because B^2 - AC > 0