CALCULUS Understanding Its Concepts and Methods

Tests for extreme values

The function
*f*
has a local maximum at a point
*c*
if
*f*(*c*) ≥ *f*(*x*)
for every point
*x*
that is sufficiently near
*c*.

The function
*f*
has a local minimum at a point
*c*
if
*f*(*c*) ≤ *f*(*x*)
for every point
*x*
that is sufficiently near
*c*.

Second derivative
testLet
*f*
be a function such that
*f'*(*c*) = 0
and the second derivative exists for
*x*
near
*c*.

If
*f''*(*c*) < 0,
then
*f*(*c*)
is a local maximum.

If
*f''*(*c*) > 0,
then
*f*(*c*) is a local minimum.

If
*f''*(*c*) = 0,
then the test fails.

The function
*f*(*x*,*y*)
has a local maximum at
(*a*,*b*)
if
*f*(*a*,*b*) ≥ *f*(*x*,*y*)
for all points
(*x*,*y*)
sufficiently close to
(*a*,*b*).

The function
*f*(*x*,*y*)
has a local minimum at
(*a*,*b*)
if
*f*(*a*,*b*) ≤ *f*(*x*,*y*)
for all points
(*x*,*y*)
sufficiently close to
(*a*,*b*).

Test for
extremaAssume
*f*(*x*,*y*)
has continuous second partial derivatives at
(*a*,*b*)
and suppose
*f _{x}*(

Let *D* = *f _{xx}*(

a
local maximum if
*D* > 0
and
* f** _{xx}*(

a
local minimum if
*D* > 0
and
* f** _{xx}*(

a
saddle point if
*D* < 0.

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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.