Endpoint and Absolute Extreme Values endpoint endpoint maximum minimum I Figure 4.4.1 Figure 4.4.2 endpoint endpoint minimum maximum Figure 4.4.3 Figure 4.4.4 Endpoint extreme values (4.4.1),p.174-175,figures 4.4.1-4 Salas,Hille,Etgen Calculus:One and Several Vartsbles Main Menu Copy right 2007 John Wiley Sons,Ine.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Endpoint and Absolute Extreme Values Endpoint extreme values (4.4.1), p. 174–175, figures 4.4.1-4
DEFINITION 4.4.1 ENDPOINT EXTREME VALUES If c is an endpoint of the domain off,then f is said to have an endpoint maximum at c provided that fc)≥fc) for allx in the domain of f sufficiently close to c. It is said to have an endpoint minimum atc provided that fc)≤fx) for allx in the domain of f sufficiently close to c. Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Endpoint and Absolute Extreme Values DEFINITION 4.4.2 ABSOLUTE EXTREME VALUES The function f is said to have an absolute maximum at d provided that f(d)≥fx) for all x in the domain of f; f is said to have an absolute minimum at d provided that f(d)≤f(x) for all x in the domain of f. Absolute extreme values (4.4.2),p.175 Saks,Hille,Etgen Calculus One and Several Varibles Main Meny Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Endpoint and Absolute Extreme Values Absolute extreme values (4.4.2), p. 175
Endpoint and Absolute Extreme Values Example 1 Find the critical points of the function fx)=1+4x2-2x4,x∈[-1,3]. Then find and classify all the extreme value. Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Endpoint and Absolute Extreme Values Example 1 Find the critical points of the function Then find and classify all the extreme value
y 9 9-2 -1 1 2 3 -名 Figure 4.4.5 Sals,Hille,Etgen Calculus One and Several Varibles Main Menu Copy right 2007 John Wiley Sons,Ine.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Endpoint and Absolute Extreme Values Example 2 Find the critical points of the function m-t轻 -3≤x<0 0≤x≤2. Then find and classify all the extreme value. Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Endpoint and Absolute Extreme Values Example 2 Find the critical points of the function Then find and classify all the extreme value
y (0,2) (2,2) (1,1) 1 2 Figure 4.4.6 Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Endpoint and Absolute Extreme Values Example 3 (a)Asx→oo,3x4-100x3+2x-5→o since3x4→o. b)Asx→-0o,5x3+12x2+80→-0 since5x3→-oo. Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sons,Ine.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Endpoint and Absolute Extreme Values
Endpoint and Absolute Extreme Values Example 4 Find the critical points of the function f(x)=6√x-x√E. Then find and classify all the extreme value. Sals,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sons,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Endpoint and Absolute Extreme Values Example 4 Find the critical points of the function Then find and classify all the extreme value
y (2,4V2) 2 6 Figure 4.4.7 Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sons,Ine.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved