# Analytically Determining Extreme Values for Functions of Several Variables

We will now look at an analytical method for determining if a critical point $(a, b)$ of $z = f(x, y)$ produces a local maximum or local minimum value, or a saddle point. In doing so, consider the difference:

(1)- If $\Delta f > 0$ for small values of $h$ and $k$ then $f$ will have a local minimum at $(a, b)$.

- If $\Delta f < 0$ for small values of $h$ and $k$ then $f$ will have a local maximum at $(a, b)$.

- If $\Delta f > 0$ for some points $(h, k)$ near $(a,b)$ and $\Delta f < 0$ for some points $(h, k)$ near $(a, b)$, then $f$ will have a saddle point at $(a, b)$.

## Example 1

**Show that the critical points of the function $f(x, y) = x^3 + y^3 - 3xy$ are $(0, 0)$ and $(1, 1)$ and analytically show that $(0, 0)$ is a saddle point of $f$.**

We must first determine the critical points of $f$. To do so, let's find the gradient of $f$ and then set it equal to the zero vector. We have that:

(2)Setting the gradient of $f$ equal to the zero vector and we have that:

(3)We thus obtain the following system of equations:

(4)From the equation above we can plug in $y = x^2$ to $x = y^2$ to get:

(5)Thus $x = 0$ or $x = 1$. These values of $x$ correspond to the critical points $(0, 0)$ and $(1, 1)$. We are now ready to analytically determine whether these points are local extrema or not.

First consider the critical point $(0, 0)$ and the difference $\Delta f = f(0 + h, 0 + k) - f(0,0)$:

(6)Note that $f(h, 0) = h^3 > 0$ for small positive $h$ and $f(h, 0) = h^3 < 0$ for small negative $h$. Thus the point $(0, 0)$ is a saddle point of $f$.