For fractions of one variable, you are able to determine special points like maximum, minimum, and inflection points.
The critical points are those where the first derivative of the function is equal to 0. For example:
In this graph we see a minimum, maximum and an inflection point in the origin. If look at the graph in the original we see that the slope of the tangent line at the origin is not equal to 0, so the derivative at that point is not equal to 0, so the origin is not a critical point. In the following graph, the origin is a critical point:
Now let’s turn to functions of 2 variables. Let’s look at the following mathematical figure. It’s a graph of a function $f(x,y)$ of two variables $x,y$:
As you can see, there are 3 peaks in this figure. In those points, the value of the function $f$ is larger than the nearby values of $f$. These are points where the function is said to have a local maximum. The larger of these 3 is called the absolute maximum. Likewise, there are 2 local minimum, of which the smaller is called the absolute minimum.
In the case of a function of one variable, this means that the tangent line at a point is an horizontal line (with slope 0), meaning that the derivative at that point is equal to 0 (if the derivative exists).
In the case of a function of two variables, if $f$ has a local maximum or minimum at a point $(a,b)$ and the first order partial derivatives exist at that point, then the first order partial derivatives at that point are equal to 0. In a formula: $f_x(a,b)=0$ and $f_y(a,b)=0$
But, is this also true the other way around? If $f_x(a,b)=0$ and $f_y(a,b)=0$, then can you conclude that $(a,b)$ is a minimum or maximum?
Check the following graph, of the function$z^2=x^2-y^2$:
The origin looks like a special point, but is not a maximum nor a minimum.
Let’s compute the partial derivatives in the origin:
But, the origin is obviously not a extreme point. This point is known as a saddle point.