What Happens at Maxima and Minima?

How can we identify points in the domain of P(x,y) which might be maxima or minima? If you think back to one variable calculus, you'll recall that the maxima occurred anywhere that the function was flat. The function is neither increasing or decreasing at this point, so it's derivative is zero. Now we have more than one direction in which the function can change. We'd really like for the function P(x,y) to be flat in all directions.

To be flat in every direction means that the direction derivative of the function must be zero in every direction. Thus, for every vector $\vec{u}$which has magnitude 1, we want

$$0 = f_{\vec{u}}(x,y) = \mbox{grad} f(x,y) \cdot \vec{u} = \v... ... \theta = \vert\vert \mbox{grad} f(x,y) \vert\vert \cos \theta.$$ (1)

This implies that either $\cos \theta = 0$ (not allowed since this limits the possible vectors $\vec{u}$) or $\vert\vert \mbox{grad} f(x,y) \vert\vert = 0$. But this second condition can only occur if the gradient of f is, in fact, zero. So, a necessary condition for the point (x₀, y₀) to be a maxima or minima of f(x,y) is that

$$\mbox{grad} f(x_0, y_0) = f_x(x_0, y_0) \hat{i} + f_y (x_0, y_0)\hat{j} = \vec{0}.$$ (2)

Thus, we have two equations

$$\frac{\partial f}{\partial x}(x_0, y_0) = 0 \qquad \qquad \frac{\partial f}{\partial y}(x_0, y_0) = 0$$ (3)

for the two unknowns x₀ and y₀.

Also note that these conditions are merely necessary, but not sufficient. Just as in one variable calculus, where inflection points creep in, here we have saddle points. At these points, the function is concave up in one direction and concave down in another. As an example, consider the origin for g(x,y) = x² - y².