Finding extrema, non-critical points, and saddle points given a contour plot

Question

I'm trying to interpret the contour plot as like a mountain, where all the segregated areas are plains that are higher when it proceeds inwards. What is the significance of the numbers on x-axis and y-axis, and wouldn't the points that are most centered be maximums?

$\hskip 12 in$

Answer 1

The general idea is:

1. When a point is surrounded by concentric closed contour lines, it have to be a extrema, because as you cross the contour lines, the height is continuously changing, and at the center you will get the zero change.

2. When two contour lines cross their is a saddle point.

3. Neither of two stated happens, then it is non-critical point.

So here, points B, D are extrema, A is saddle point, and C is non-critical point.

Answer 2

There are incorrect statements in the accepted answer, and there are some subtleties not mentioned.

1. Concentric closed contour lines always indicate either a local minimum or a local maximum.
2. For a well-defined function, two contour lines with different values can never intersect. Since contour lines indicate function values, if two different contour lines were to intersect, the function would have two different values at that point, which is impossible. So a function is undefined at a point where two different contour lines intersect.
3. If a contour line intersects itself, the point could be a saddle point, local minimum, or local maximum. For example, $f(x,y) = x y$ has a saddle point at the origin where the $0$-contour line intersects itself, $f(x,y) = x^2 y^2$ has a local minimum at the origin where the $0$-contour line intersects itself, and $f(x,y) = -x^2 y^2$ has a local maximum at the origin where the $0$-contour line intersects itself.
4. Without the values of the contour lines, nothing can be said about a point of intersection of contour lines. It could be a point at which the function is undefined, a saddle point, or a local extrema.

To answer the question, it appears that points $B$ and $D$ are local extrema, which could be minima or maxima. The function $C$ is not a critical point. Not much can be said about the point $A$ - if the values of the contour lines that are tangent at $A$ are different, then point $A$ looks like a point where the slope becomes infinite. On the other hand, if the values are the same, then the point $A$ could be a local minimum, local maximum, or a saddle point.

A possible function which would generate the contour plot (shown on the left below) in the question is $$ f(x,y) = ((x-1)^2 + y^2 - 1) \left( \left(x - \frac{3}{2} \right)^2 + y^2 - \frac{9}{4} \right) , $$ which will have a zero set consisting of a circle of radius $1$ centered at $(1,0)$ and a circle of radius $3/2$ centered at $(3/2,0)$. Thus, the 0-contours will be tangent at $(0,0)$, where there is a critical point. In this case, there is a saddle point at $(0,0)$: there is a path convergint to $(0,0)$ consisting of points outside the circle $(x-1)^2 + y^2 = 1$ and inside the circle $(x-3/2)^2 + y^2 = 9/4$ along which $f(x,y) < 0$, and also a path along the $x$-axis converging to $(0,0)$ along which $f(x,y) > 0$.

However, the function $g(x,y) = f(x,y)^2$ will have a similar contour plot (shown to the right below) - in particular the same zero set - and still a critical point at $(0,0)$, but is everywhere positive, and so the point $(0,0)$ where the contour lines are tangent is a local minimum.