Three plots of this function appear on p. 636 of the text. Here are the Maple equivalents (see also the worksheet for Section 15.1).

> plot3d( f, x=-5..5, y=-5..5, orientation=[-45,45], title="Figure 16 a" );

> plot3d( f, x=-5..5, y=-5..5, style=patchcontour, orientation=[-45,45], title="Figure 16 b" );

> plot3d( f, x=-5..5, y=-5..5, style=contour, orientation=[-90,0],
title="Figure 16 c" );

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From these pictures it is clear that there is a unique global maximum and minimum for this function. The critical points appear to have y=0 and the maximum occurs with x>0 while the minimum occurs with x<0.

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New, let's try to locate these point via calculus.

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> fx := diff( f, x );

> fy := diff( f, y );

> fy := simplify( fy );

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> solve( {fx=0, fy=0}, {x,y} );

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Thus, Maple finds two critical points. To classify these points as local extrema, use the Second Derivative Test:

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> fxx := diff( f, x,x );

> fyy := diff( f, y,y );

> fxy := diff( f, x,y );

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> discr := simplify( fxx*fyy-fxy^2 );

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One critical point is (0,1):

> cp1 := {x=0,y=1};

> eval( discr, cp1 );

> eval( fxx, cp1 );

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With D>0 and fxx<0, this critical point is a local maximum.

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The second critical point is (0,-1):

> cp2 := {x=0,y=-1};

> eval( discr, cp2 );

> eval( fxx, cp2 );

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With D>0 and fxx>0, this critical point is a local maximum.

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