# Thread: Finding global max and min of a function

1. ## Finding global max and min of a function

I need to find the global maximum and minimum of the function
(3x^2+2xy+y^2)/(x^2-2xy+3y^2).
I did the partial derivatives, set them equal to zero, took the numerator since the function will only equal zero when the numerator is zero, and got the following:
y(-x^2+2xy+y^2) = 0 and 0 = x(x^2-2xy-y^2)

I know that (x,y) can't equal (0,0) because the function is undefined there, so I need to find other (x,y) so that -x^2+2xy+y^2 = 0 and x^2-2xy-y^2 = 0. I recognized that those are both quadratic equations, so I used the quadratic formula to find y in terms of x and got
y = (-1(+/-)2^.5)x

I don't know what to do from there. How do I find the critical points now that I have that equation for y?

Thanks!

2. ## Re: Finding global max and min of a function

Originally Posted by tubetess123
I need to find the global maximum and minimum of the function
(3x^2+2xy+y^2)/(x^2-2xy+3y^2).
I did the partial derivatives, set them equal to zero, took the numerator since the function will only equal zero when the numerator is zero, and got the following:
y(-x^2+2xy+y^2) = 0 and 0 = x(x^2-2xy-y^2)

I know that (x,y) can't equal (0,0) because the function is undefined there, so I need to find other (x,y) so that -x^2+2xy+y^2 = 0 and x^2-2xy-y^2 = 0. I recognized that those are both quadratic equations, so I used the quadratic formula to find y in terms of x and got
y = (-1(+/-)2^.5)x

I don't know what to do from there. How do I find the critical points now that I have that equation for y?

Thanks!
Finding Critical Points:

Find derivative of function, set the deriv equal to zero and solve for x.

example:

f(x) = x^3 - 3x^2 + 13
f'(x) = 3x^2 - 6x

3x^2 - 6x = 0
3x(x-2) = 0
x = 0, 2 <--These are your critical points.

Plug the critical values 0 and 2 into the original equation.
(0, 13) global max
(2,9) global min

3. ## Re: Finding global max and min of a function

I know how to find the CP of a single-variable function... A multivariable function is different and more complex, and in the case I provided I am extremely confused as to what I am to do to find the actual coordinates of the critical points.