Exercise · ZW-ABSOLUTE-EXTREMA-ON-A-REGION-004 · ★☆☆ Basic
What condition guarantees that \(f\) attains an absolute maximum and minimum on a region \(D\)?
Choose one option.
Show answer
- A. That \(f\) is continuous on a closed, bounded region
Explanation
This is the extreme value theorem: a continuous function on a closed, bounded set attains its maximum and minimum. Without closedness (for example \(f=x\) on \(0<x<1\)) or without continuity, it may fail to attain them.
Review the concept: Absolute extrema on a region
The absolute extrema of a function \(f\) on a closed, bounded region \(D\) are the minimum \(m\) and the maximum \(M\) of the values \(f\) takes on \(D\).
\[m=\min_{D} f,\qquad M=\max_{D} f\]
More practice problems: absolute extrema on a region
- ZW-ABSOLUTE-EXTREMA-ON-A-REGION-001 Find the absolute maximum of f(x,y)=2x+3y on the rectangle [0,1]×[0,2].
- ZW-ABSOLUTE-EXTREMA-ON-A-REGION-002 Find the absolute minimum of f(x,y)=x^2+y^2-2x-4y+7 on the square [0,3]×[0,3].
- ZW-ABSOLUTE-EXTREMA-ON-A-REGION-003 Find the absolute maximum of the same function f(x,y)=x^2+y^2-2x-4y+7 on [0,3]×[0,3].
- ZW-ABSOLUTE-EXTREMA-ON-A-REGION-005 Find the absolute maximum of f(x,y)=xy on the triangle with vertices (0,0), (2,0), (0,2).
- ZW-ABSOLUTE-EXTREMA-ON-A-REGION-006 Find the absolute minimum of f(x,y)=x^2-y^2 on the disk x^2+y^2≤1.
- ZW-ABSOLUTE-EXTREMA-ON-A-REGION-007 Find the absolute maximum of f(x,y)=x^2+2y^2 on the disk x^2+y^2≤4.
- ZW-ABSOLUTE-EXTREMA-ON-A-REGION-008 Find the absolute minimum of f(x,y)=x^3-3x+y^2 on the rectangle [-2,2]×[-1,1].
- ZW-ABSOLUTE-EXTREMA-ON-A-REGION-009 If the absolute minimum of f on D is attained at an interior point of D, what can be said about tha…
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