# Using Function Hess To Analyze Extrema, - HP 48gII User Manual

Graphing calculator.

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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
2
2
, fXY(X,Y) = ∂
f/∂X/∂Y, and fYY(X,Y) = ∂
f/∂X
The last result indicates that the discriminant is ∆ = -12X, thus, for (X,Y) = (1,0),
∆ <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 φ(x
']. Function HESS returns the Hessian matrix of the function φ, defined
'x
'...'x
2
n
as the matrix H = [h
] = [∂
ij
to the n-variables, grad f = [ ∂φ/∂x
variables ['x
' 'x
'...'x
'].
1
2
n
2
f/∂Y
, x
, ...,x
), and a vector of the functions ['x
1
2
n
2
φ/∂x
∂x
], the gradient of the function with respect
i
j
, ∂φ/∂x
, ... ∂φ/∂x
1
2
2
.
2
2
f/∂X
<0 (relative
], and the list of
n
Page 14-6
'
1

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