Total Differential Of A Function Z = Z(X,Y); Determining Extrema In Functions Of Two Variables - HP 50g User Manual

Total differential of a function z = z(x,y)

From the last equation, if we multiply by dt, we get the total differential of the
function z = z(x,y), i.e., dz =
A different version of the chain rule applies to the case in which z = f(x,y), x =
x(u,v), y = y(u,v), so that z = f[x(u,v), y(u,v)]. The following formulas represent
the chain rule for this situation:
z
u

Determining extrema in functions of two variables

In order for the function z = f(x,y) to have an extreme point (extrema) at (x
its derivatives f/ x and f/ y must vanish at that point. These are necessary
conditions. The sufficient conditions for the function to have an extreme at point
(x ,y ) are f/ x = 0, f/ y = 0, and
o o
The point (x ,y ) is a relative maximum if
o o
f/ x > 0. The value
If = ( f/ x ) ( f/ y )-[ f/ x y]
saddle point, where the function would attain a maximum in x if we were to
hold y constant, while, at the same time, attaining a minimum if we were to
hold x constant, or vice versa.
Example 1 – Determine the extreme points (if any) of the function f(X,Y) = X
2
Y +5. First, we define the function f(X,Y), and its derivatives fX(X,Y) = f/ X,
fY(X,Y) = f/ Y. Then, we solve the equations fX(X,Y) = 0 and fY(X,Y) = 0,
simultaneously:
z/ x) dx + ( z/ y) dy.
(
z x z y
x u y u
is referred to as the discriminant.
z z
,
v x
= ( f/ x ) ( f/ y )-[ f/ x y]
f/ x < 0, or a relative minimum if
2
< 0, we have a condition known as a
x z y
v y v
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,y ),
o o
2
> 0.
3
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