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A1. Relative and Absolute Extrema

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 R2? 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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