Does the above result generalize to R

^{2}? In particular, say f is differentiable on the entire plane, and has a relative maximum at (0,0), and no other relative extrema. Must the function have an absolute maximum at (0,0)? Why or why not?

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A1. Relative and Absolute Extrema

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In finding minima and maxima, first-year calculus students often use the fact that, for a function differentiable on the entire real line, if the function has exactly one relative extremum, that extremum is an absolute extremum. The proof is simple: WLOG, say the function has a relative maximum at a. If a is not an absolute maximum, then there is a point b where f(b) > f(a). Then on the closed interval with endpoints a and b, f has a minimum value. Since the minimum is less than f(a), it is not attained at either a or b, so it is in the open interval, and thus is a relative minimum.

Does the above result generalize to R^{2}? In particular, say f is differentiable on the entire plane, and has a relative maximum at (0,0), and no other relative extrema. Must the function have an absolute maximum at (0,0)? Why or why not?

Does the above result generalize to R

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Thanks,

Peter

I will sometimes post a link to a solution. If I haven't yet posted a link, you may send me your answer and if it is correct I will credit you on the blog. To send answers, please mailto: peterash3@gmail.com.

Thanks,

Peter

From time to time post problems that are somewhat more advanced than those in the Elementary Math Problems. These problems will require a knowledge of some college-level mathematics, either for their statement or for the solution that I know. Advanced math problems will be denoted by the letter A followed by the problem number.

I will sometimes post a link to a solution. If I haven't yet posted a link, you may send me your answer and if it is correct I will credit you on the blog. To send answers, please mailto: peterash3@gmail.com.

I will sometimes post a link to a solution. If I haven't yet posted a link, you may send me your answer and if it is correct I will credit you on the blog. To send answers, please mailto: peterash3@gmail.com.

Here are some recent reviews on mathematics, learning theory, education, and related technology:

The Number Sense: How the Mind Creates Mathematics by Stanislaus Dehaene, Oxford University Press 1997

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The King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, Walker and Company, 2006

The Poincare Conjecture: In Search of the Shape of the Universe, Donal O'Shea, Walker & Company, 2007

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, George Lakoff and Rafael E. Nunez, Basic Books, 2000.

The Number Sense: How the Mind Creates Mathematics by Stanislaus Dehaene, Oxford University Press 1997

Three Books on the Riemann Hypothesis

The King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, Walker and Company, 2006

The Poincare Conjecture: In Search of the Shape of the Universe, Donal O'Shea, Walker & Company, 2007

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, George Lakoff and Rafael E. Nunez, Basic Books, 2000.

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