CriticalPoints -...

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Critical Points Warren Weckesser Here we take a look at the critical points of a function. > with(plots): Warning, the name changecoords has been redefined > f := x^3-3*x+y^3-3*y; := f - + - x 3 3 x y 3 3 y Compute the first derivatives. > fx := diff(f,x); := fx - 3 x 2 3 > fy := diff(f,y); := fy - 3 y 2 3 Let Maple find the critical points. > solve({fx=0,fy=0},{x,y}); , , , { } , = x 1 = y 1 { } , = x 1 = y -1 { } , = x -1 = y 1 { } , = x -1 = y -1 Create a contour diagram. > c := [seq(i/2,i=-10. .10)]; := c ± ² ² ³ ´ µ µ , , , , , , , , , , , , , , , , , , , , -5 -9 2 -4 -7 2 -3 -5 2 -2 -3 2 -1 -1 2 0 1 2 1 3 2 2 5 2 3 7 2 4 9 2 5 > contourplot(f,x=-2. .2,y=-2. .2,contours=c,grid=[80,80],color=black, thickness=2); > plot3d(f,x=-2. .2,y=-2.
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Unformatted text preview: .2,axes=boxed,style=patchcontour,grid=[60,60 ],contours=c,orientation=[-50,40],shading=zgrayscale); Take a closer look around x=-1, y=1. > contourplot(f,x=-2. .0,y=0. .2,grid=[80,80],contours=c,color=black,t hickness=2); > contourplot(f,x=-1.2. .-0.8,y=0.8. .1.2,grid=[80,80],contours=25,col or=black,thickness=2); Note that as we zoom in on the critical point at (-1,1), the graph looks like a saddle. > plot3d(f,x=-1.2. .-0.8,y=0.8. .1.2,grid=[30,30],shading=zgrayscale,a xes=boxed,orientation=[-35,60]); >...
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CriticalPoints -...

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