Gradient and Optimization

For a standard vector calculus on xyz-coordinate, the package "vect" is useful and it is needed to load at the beginning:

--> load(vect)

The gradient operator "grad" in Maxima must be "expressed" and their partial derivatives to be "evaluated."

--> f: 10*x^2*y - 5*x^2 - 4*y^2 - x^4 - 2*y^4
--> del_f: ev(express(grad(f)),diff)

Then we get the gradient

. The third component calculates the partial derivative

with respect to z even when we define a function of

. Then we can find all the critical points by solving the system of equations

and

--> realonly: true;
--> sol: solve([del_f[1]=0, del_f[2]=0])

We set realonly in order to produce only real-valued solutions when solve is executed. Each solution can be found by calling sol[1], sol[2], and so on. Now we need to proceed the second derivative test to determine a local minimum, a local maximum, or a saddle point for each critical point.

--> f_xx: diff(f,x,2)
--> D: diff(f,x,2)*diff(f,y,2) - diff(f,x,1,y,1)^2
--> D, sol[1]; f_xx, sol[1];
--> D, sol[2]; f_xx, sol[2];
--> D, sol[3]; f_xx, sol[3];
--> D, sol[4]; f_xx, sol[4];
--> D, sol[5]; f_xx, sol[5];

Here the variables x and y are substituted by each solution.

Similarly we can solve the optimization problem with constraint .

--> f: x*y*z;
--> g: 2*x*z + 2*y*z + x*y
--> del_f: ev(express(grad(f)),diff)
--> del_g: ev(express(grad(g)),diff)

By applying the method of Lagrange multipliers, we can find the solutions:

--> solve([del_f[1]=lmd*del_g[1], del_f[2]=lmd*del_g[2], del_f[3]=lmd*del_g[3], g=12])

You may use wxMaxima and complete some of the even-numbered exercises assigned in the class.

section14-6.wxm and exercise14-6.wxm (the directional derivative and the tangent plane)
section14-7.wxm, section14-8.wxm, and exercise14-78.wxm (the second derivative test and the method of Lagrange multipliers)