(819,#17) Use a graph and/or level curves to estimate the local maximum and minimum values and saddle points of the function . Then use calculus to find these values precisely.

Solution: A graph on the domain x = -2 ..2 to y = -1.. 3 shows a possible maximum and a possible minimum on the line x = 0. The saddle points are not too clear.

So we make a contour plot for x = -1..1 and y = -0.5 .. 2.5. It shows maxima at (0,0) and (0,2).

Contour plots centered at (1,1) and (-1,1) indicate likely saddles.

So we go on to the analysis, using calculus

To find the critical points, solve the equations:

Substituting the values x = 0, y = 1 into :

Therefore, the critical points are

Using the Second Derivative Test,

So, f(0,0) = 2 is a local maximum, f(0,2) = -2 is a local minimum and (1,1) (-1,1) are saddle points.

Maple code

by: sh