__Lagrange Multiplier Experiments with Level Curves__

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**with(linalg): with(plots):**

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**Lagrange Multipliers and Level Curves**

**We want to find the max and min of f(x,y)=x^3+x*y+2*y^3 on the constraint set**

**g(x,y)=x^2+x*y+y^2-1=0. **

**First below is a picture of the constraint set. **

**Remember we **
**only**** care about the values of f(x,y) on the ellipse pictured below.**

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**g:=x^2+x*y+y^2-1;
gr0:=implicitplot(g=0,x=-2..2,y=-2..2,thickness=2,color=black):
display(gr0);**

**Can you make a rough guess about where on the above ellipse x^3+x*y+2*y^3 would be biggest?**

**(e.g. Which quadrant? Closer to the x-axis or y-axis? About what might the maximum value be?)**

**Below are some level curves of f(x,y)=x^3+x*y+2*y^3 superimposed.**

**Look near the point (0,1) on the top level curve where f(x,y) is 2.
If you move a little to the right of (0,1) **

How about if you move a little to the left of (0,1)?

**What does this say about the possibility that (1,0) might be a **
**constrained local maximum**** of f restricted to the ellipse?**

**Roughly which way are the gradients of f and g pointing at (1,0)? (You can read this from the picture...)**

**Could they be parallel at that point?**

**Harder but the key to Lagrange Multipliers: **

** Can you link the non-parallelism of the gradients at (1,0) to (1,0) **
**not ****being a constrained local maximum?**

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**f:=x^3+x*y+2*y^3;
gr1:=contourplot(f,x=-2..2,y=-2..2,contours=[-2,1.5,2],coloring=[red,green,blue],thickness=2):
display([gr0,gr1],scaling=constrained);**

**Try and adjust the contours values (you can add more), so that one of the level curves is tangent to the ellipse.**

**You can click on the tangency spot with your mouse and look in the upper left to read the point (x,y) more accurately.**

**Or select Plot Display->Windows from the Options menu for bigger plots.**

**How many level curves with points of tangency can you find? (We found at least 4...)**

**From the Lagrange Multipliers point of view, **
**each of these ****is a possible constrained local minimum or maximum.**

**What do you think are the biggest and smallest values of f on the ellipse?**

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**gr2:=contourplot(f,x=-2..2,y=-2..2,contours=[-2,1.5,2],coloring=[red,green,blue,violet],thickness=2):
display([gr0,gr2],scaling=constrained);**

Partial Derivatives.

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**f_x:=diff(f,x);
f_y:=diff(f,y);
g_x:=diff(g,x);
g_y:=diff(g,y);**

**You can use numerical solution routines to search for solutions.**

**(The second form of the command seeks solutions in a certain range.)**

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**sol1:=fsolve({f_x=lambda*g_x,f_y=lambda*g_y,g=0},{x,y,lambda});
sol2:=fsolve({f_x=lambda*g_x,f_y=lambda*g_y,g=0},{x,y,lambda},{x=-0.5 .. 0.5,y=0..2});**

**These lines evaluate f at the points found above. **

**Can you compare these to the values of f you found for the level curves which were tangent?**

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**subs(sol1,f); subs(sol2,f);**

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