Chapter 3-Higher Order Derivatives

Higher Order Derivatives :




3 1 .
:
Iterated Partial Derivatives


Of :
of Equality of Partial Derivatives
Oxdy dydx



of (o Yo) 1 (S(DX DY)
WM
:
,
and Xo + DX
between Xo
,


OXdY BXDY eX
S(DX DY)
(1x Yo + DY)
- Yo
:
, , ,




= of (x , Y) DXDY
dydX



3 2 .
:
Taylor's Theorem


f( % o + h) =
f(Yo) +
hio) + Ri , ) First-order Taylor's Theorem

R
, (o 2) ,
=
+(xo n)-, f(xo)-(Df(o)](n)

↑ o + h) =
/Ito) +
Shilhini(o) + Ro Second-order Taylor's There is


account for
of, ofoa

3
.3 :
Extrema of Real-valued Functions


First-Derivative Test :
Df(o) 0 =
shows critical points




Second-Derivative Test :




Hessian : HEYolIn) : Tox Moshin ; matrix
esymmetric

I Moxi.... Ooo()
ha]
-
h, ...
flox
:

· flexnox

=
(1) (IXn](nxn](nx1] :
[17 Scalar output


Hf(Yo)(n) >0 >
-
minimum



HF(X)(5) <0 >
-
maximum



2-variables :




minimum :




I .




(Xo , Yo)
= ( Y ,




2 .




Of(X Yo ,




3
. D
( )-l0@(0 40
=
,




maximum :




(o Yo
= ( Y.
1 .

, ,




2 .




Of(X You ,




.
3 D
( )-l0@(0 40
=
,