Exercise · ZW-ABSOLUTE-EXTREMA-ON-A-REGION-002 · ★☆☆ Basic
Find the absolute minimum of \(f(x,y)=x^2+y^2-2x-4y+7\) on the square \([0,3]\times[0,3]\).
Choose one option.
Show answer
- B. \(2\)
Explanation
The derivatives \(f_x=2x-2\), \(f_y=2y-4\) vanish at \((1,2)\), which lies inside the square, and \(f(1,2)=2\). On the boundary \(f\) takes larger values (for instance \(f(0,2)=3\)), so the absolute minimum is \(2\).
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-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?
- 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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