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Differentiation

Differentiation Functions

Saddle Points

A function

has a saddle point at

if

and has both positive and negative values around

 > Saddle Points

Local Min/Max

latex!encoded:base64,Zih4KVxlcHNpbG9uIFw7Q157bn0sYVxsZXEgeFxsZXEgYg==

where

latex!encoded:base64,Q157bn0=

means

latex!encoded:base64,bnRo

order continuous

If

latex!encoded:base64,ZicoeCk=

changes from positive to negative at

latex!encoded:base64,eF97MX0=

, then

latex!encoded:base64,Zih4X3sxfSk=

is a local max

If

latex!encoded:base64,ZicoeCk=

changes from negative to positive at

latex!encoded:base64,eF97MX0=

, then

latex!encoded:base64,Zih4X3sxfSk=

is a local min

 > Local Min/Max

Rolle’s Theorem

If

and

,

then there exists a number

,

such that

 > Rolle’s Theorem

Lagrange Multipliers

Maximize:

subject to

Let:

Solve the system of equations:

Enter solved values of

into

and identify min and max.

Make sure solutions make sense.

 > Lagrange Multipliers

Jacobian

latex!encoded:base64,Sj1cYmVnaW57Ym1hdHJpeH0KXGZyYWN7XHBhcnRpYWwgeV8xfXtccGFydGlhbCB4XzF9ICZcY2RvdHMgICZcZnJhY3tccGFydGlhbCB5XzF9e1xwYXJ0aWFsIHhfbn0gXFwgCiBcdmRvdHMmXGRkb3RzICAgJlx2ZG90cyBcXCAKIFxmcmFje1xwYXJ0aWFsIHlfbX17XHBhcnRpYWwgeF8xfSZcY2RvdHMgICYgXGZyYWN7XHBhcnRpYWwgeV9tfXtccGFydGlhbCB4X259ClxlbmR7Ym1hdHJpeH0=
 > Jacobian
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