Exercise · ZW-ABSOLUTE-EXTREMA-ON-A-REGION-001 · ★☆☆ Basic
Find the absolute maximum of \(f(x,y)=2x+3y\) on the rectangle \([0,1]\times[0,2]\).
Choose one option.
Show answer
- C. \(8\)
Explanation
Since \(\nabla f=(2,3)\) never vanishes, there are no interior critical points; the maximum lies on the boundary. Because \(f\) increases in both \(x\) and \(y\), it occurs at the corner \((1,2)\): \(f=2+6=8\).
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
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- 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-004 What condition guarantees that f attains an absolute maximum and minimum on a region D?
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- 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].
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