Exercise · ZW-ABSOLUTE-EXTREMA-ON-A-REGION-009 · ★★☆ Intermediate
If the absolute minimum of \(f\) on \(D\) is attained at an interior point of \(D\), what can be said about that point?
Choose one option.
A It is necessarily a boundary point B It must be a point where the second derivative test gives a local minimum C It is a critical point of \(f\), or a partial derivative fails to exist there D The gradient is nonzero and points outward Show answer C. It is a critical point of \(f\), or a partial derivative fails to exist there← Back to the question
Explanation In the interior, an absolute minimum is also a local minimum , so the gradient is zero or does not exist (a critical point ). The converse fails, and the minimum may also lie on the boundary.
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].