The Second Derivative Test

The second derivative test for functions of 2 or more variables is an extremely important tool that is used throughout mathematics and science. In this worksheet, we will explore several of these applications and will also extend the second derivative test to higher dimensional settings. First, we load some packages.

> restart:with(linalg):with(plots):with(plottools):with(student):

# matsubs(eqs,matexpr) uses the list of equations eqs to
# substitute into the coefficients of the matrix matexpr

matsubs:=proc(eqs::list,matexpr::matrix)
local i,j,ii,jj,matexpn;
ii:=rowdim(matexpr);
jj:=coldim(matexpr);
matexpn := matrix(ii,jj);
for i from 1 to rowdim(matexpr) do
for j from 1 to coldim(matexpr) do
matexpn[i,j] := subs(eqs,matexpr[i,j]);
end do
end do;
evalm(matexpn)
end proc:

>

Critical Points, Saddles, and Extrema

Least Squares

Optimization with more than 2 Variables

Exercises