Using Maple to plot the surface f(x,y) = 2 x3 + x y2 + 5 ...
Using Maple to plot the surface f(x,y) = 2 x3 + x y2 + 5 x2 + y2
By solving fx = 6x2 + y2 + 10 x = 0 and fy = 2xy + 2 y = 0, we can determine that the critical points were (0, 0), (-5/3, 0), (-1, -2) and (-1, 2). By calculating D= fxx fyy – fxy2 = (12x+10)(2x+2) – (2y)2 we can determine that
At (0,0) D = 20 > 0 and fxx = 10 > 0 which implies (0,0) is a local min
At (-5/3,0) D = 40 / 3 > 0 and fxx = -10 < 0 which implies that (-5/3,0) is a local max
At (-1,-2) and (-1,2) D = -16 which implies that these points are saddle points.
To plot the surface in Maple we can use the code:
> x:=(u,v)->u:
> y:=(u,v)->v:
> z:=(u,v)-> 2*u^3+u*v^2+5*u^2+v^2:
> plot3d([x,y,z], -2.5..1, -5..5, style=PATCH, numpoints=1000, axes=BOX,
view=-0.5..5, labels=[`x`,`y`,`z`]);
Here “–2.5..1” limits x to be in –2.5 ( x ( 1, “-5..5” limits y to be in –5 ( y ( 5 and “view = -0.5..5” limits z to be in –0.5 ( z ( 5. I tried various values for these limits until I got a picture that would show the peaks and valleys. Also the commands style, numpoints, axes and labels control how the graph looks and is labeled. In Maple type “help(numpoints);” etc. to get help on these.
To draw a contour plot in maple we can use
> with(plots):
> contourplot([x,y,z],-2.5..1,-5..5,contours=100);
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