Exercise · ZW-ABSOLUTE-EXTREMA-ON-A-REGION-010 · ★★★ Advanced
Find the absolute maximum of \(f(x,y)=x^2y\) on the triangle \(x\ge0\), \(y\ge0\), \(x+y\le3\).
Choose one option.
Show answer
- C. \(4\)
Explanation
In the interior \(f_x=2xy\), \(f_y=x^2\) vanish only along \(x=0\), where \(f=0\). On the legs \(f=0\). On the hypotenuse \(y=3-x\): \(f=x^2(3-x)\), with derivative \(6x-3x^2=0\) at \(x=2\), and \(f=4\). The absolute maximum is \(4\), at \((2,1)\).
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-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].