# Thread: Extreme values of multivariable function

1. ## Extreme values of multivariable function

Hello,

Here's the problem:

Find the highest and lowest points on the curve x^2+xy+y^2

What do I do here? I used implicit differentiation to find dy/dx, but I'm not sure where to go from there or if that even makes sense. Do that standard rules applies for the first derivative test? Any help would be greatly appreciated.

2. Originally Posted by riverjib
Hello,

Here's the problem:

Find the highest and lowest points on the curve x^2+xy+y^2

What do I do here? I used implicit differentiation to find dy/dx, but I'm not sure where to go from there or if that even makes sense. Do that standard rules applies for the first derivative test? Any help would be greatly appreciated.
You need to find the relative extrema.

$\displaystyle f(x,y)=x^2+xy+y^2$
Then,
$\displaystyle \frac{\partial f}{\partial x}=2x+y$
$\displaystyle \frac{\partial f}{\partial y}=2y+x$
We require that,
$\displaystyle 2y+x=0$
$\displaystyle 2x+y=0$
Thus,
$\displaystyle (x,y)=(0,0)$

To check what this is we use the second partials test.
$\displaystyle f_{xx}=2$
$\displaystyle f_{yy}=2$
And, because of continuity,
$\displaystyle f_{xy}=f_{yx}=0$
Thus,
$\displaystyle f_{xx}f_{yy}-[f_{xy}]^2 =4-0>0$
Thus, the test works.

Now we check the sign of either $\displaystyle f_{xx}(0,0)$ or $\displaystyle f_{yy}(0,0)$. Which are positive. Thus the point is a relative minima.

(Not necessarily a global minima).