Absolute Extreme Values DEFINITION 16.6.1 ABSOLUTE EXTREME VALUES Suppose that f is a function of several variables. f is said to have an absolute maximum at xo provided that f(xo)=f(x)for all x in the domain of f; f is said to have an absolute minimum at xo provided that fxo)≤f(x) for all x in the domain of f. Main Meny cmew5时
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Absolute Extreme Values
Absolute Extreme Values DEFINITION 16.6.2 A subset S of the plane or three-space is said to be bounded provided there exists a positive number R such that Ix≤R for all x∈S. THEOREM 16.6.3 THE EXTREME-VALUE THEOREM If f is continuous on a closed and bounded set D.then on that set f takes on an absolute maximum and an absolute minimum Main Menu C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Absolute Extreme Values
Example 1.Find the absolute extreme values taken on the function f(x,)=x2+2-2x-2+4 on the closed disk D={(,y:2+2<9). Main Meny o墙8的
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Example 2.Find the absolute extreme values of the function fc,)=4ry-x2-2-6z on the triangular region D={(x,y):0≤x≤2,0≤y≤3x}. Main Menu
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Example 3.Consider the function f(x,)=1+V/x2+y2 on the triangular region D={(x,y):0≤x≤a,0≤y≤b} Main Meny C 007e
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11 Example 4.f(x,)=xy++- x>0,y>0. x y Main Meny c8的
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