Exercise · ZW-ABSOLUTE-EXTREMA-ON-A-REGION-008 · ★★☆ Intermediate
Find the absolute minimum of \(f(x,y)=x^3-3x+y^2\) on the rectangle \([-2,2]\times[-1,1]\).
Choose one option.
Show answer
- B. \(-2\)
Explanation
Critical points: \(f_x=3x^2-3=0\), \(f_y=2y=0\) give \((\pm1,0)\) with \(f(1,0)=-2\) and \(f(-1,0)=2\). On the boundary, \(f(-2,0)=-2\) and the other candidates are larger. The absolute minimum is \(-2\); it is attained at \((1,0)\) and at \((-2,0)\).
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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