Using Function Hess To Analyze Extrema - HP 50g User Manual

Graphing calculator.

We find critical points at (X,Y) = (1,0), and (X,Y) = (-1,0). To calculate the
discriminant, we proceed to calculate the second derivatives, fXX(X,Y) =
2
X
, fXY(X,Y) =
The last result indicates that the discriminant is
<0 (saddle point), and for (X,Y) = (-1,0), >0 and
maximum). The figure below, produced in the calculator, and edited in the
computer, illustrates the existence of these two points:

Using function HESS to analyze extrema

Function HESS can be used to analyze extrema of a function of two variables as
shown next. Function HESS, in general, takes as input a function of n
independent variables (x
'x
'...'x
']. Function HESS returns the Hessian matrix of the function , defined
2
n
as the matrix H = [h
the n-variables, grad f = [
variables ['x
' 'x
1
f/ X/ Y, and fYY(X,Y) =
, x
, ...,x
1
2
2
] = [
/ x
ij
i
/ x
'...'x
'].
2
n
f/ Y
= -12X, thus, for (X,Y) = (1,0),
), and a vector of the functions ['x
n
x
], the gradient of the function with respect to
j
,
/ x
, ...
1
2
2
.
2
f/ X
<0 (relative
/ x
], and the list of
n
f/
'
1
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