Exercise · ZW-ABSOLUTE-EXTREMA-ON-A-REGION-006 · ★★☆ Intermediate
Find the absolute minimum of \(f(x,y)=x^2-y^2\) on the disk \(x^2+y^2\le1\).
Choose one option.
Show answer
- D. \(-1\)
Explanation
The only critical point is \((0,0)\) (a saddle, \(f=0\)). On the boundary \(x=\cos t\), \(y=\sin t\) and \(f=\cos 2t\), which ranges from \(-1\) to \(1\). The absolute minimum is \(-1\), at \((0,\pm1)\); a critical point need not be an extremum.
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-007 Find the absolute maximum of f(x,y)=x^2+2y^2 on the disk x^2+y^2≤4.
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- 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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