What are the minimum and maximum values?

Use a graph or level curves or both to find the local maximum and minimum values and saddle points of the function.?

• Use a graph or level curves or both to find the local maximum and minimum values and saddle points of the function. Then use calculus to find these values precisely. (Enter your ...show more

• Answer:

  Obviously f(0,0) = 0, and in fact f is zero all the way along each of the two coordinate axes. You can see that the value of the factor e^(-x^2-y^2) gets smaller and smaller as you go away from the origin; for reasons of symmetry, I would expect a pair of maxima along the line y=x and a pair of minima along the line y = -x. So I'm going to consider the function f(x,x) = 3x^2 e^(-2x^2). Now I'm going to fooplot.com to graph y = 3x^2 e^(-2x^2), and it appears that the maximum occurs around x = 0.7. So the 4 extrema of f are near (0.7,0.7, 0.55), (0.7,-0.7,- 0.55), (-0.7,0.7,-0.55) and (-0.7,-0.7,0.55), if you think of the f-value as the z-coordinate. Now to the calculus. A very VERY simple bit of calculus will improve the above answer. Since f(x,x) = 3x^2 * e^(-2x^2), let's look at the derivative of that expression with respect to x; it comes out to 6x * e^(-2x^2) - 12x^3 * e^(-2x^2). This will be zero when x^2 = 1/2, i.e., when x = 0.7071. So the visual approximation of x = 0.7 was pretty good. For example, the maximum point in the first quadrant is at (sqrt(0.5), sqrt(0.5),3/(2e)), or about (0.7071, 0.7071, 0.5518). The extrema in the other 3 quadrants are obtained from this by just changing some signs: the sign of xy*f(x,y) is positive in all cases.