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ExtremumOverSet

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Description

Summary

The largest or smallest value an expression takes over a set, and the points where it
takes it: max(f(t), t in S), min, argmax, argmin.

Remarks

Over a finite set of numbers the expression is evaluated at every element and the values
compared, which is the definition. Over a closed interval with numeric ends the extremum is
attained at an endpoint or at a stationary point inside, so the candidates are the closed
endpoints and the zeros of the derivative in the open interval, which the solver is asked
for; a periodic family of zeros -- pi/6 + 2 pi n -- is read as a line in its
parameter and the whole numbers that land inside the interval are enumerated. An open
endpoint is not a candidate, since a value there is not attained: max(x, x in [0; 1)) has no maximum, and is left as written rather than answered 1.
 Two guards, because a wrong maximum is worse than none. The expression has to be one
that is smooth on the reals -- sums, products, non-negative whole powers, sines, cosines,
and exponentials with a positive base -- so that every interior extremum is a zero of the
derivative; abs(x) has its minimum where its derivative is undefined, and
1/x has no maximum on [-1; 1] at all. And the candidates' best value is
checked against the expression sampled along the interval: the solver's list of zeros is
not guaranteed complete, and a sample that beats the best candidate means one was missed,
on which the question is left as written. A sample can miss a narrow peak, so this is a
guard and not a proof; what it rules out is the wrong answer the incomplete list would
otherwise give with confidence.
 The value is returned symbolically -- the expression at the best point, simplified -- and
the points as the candidates themselves, so max(sin(t)^3 cos(t), t in [0; pi/2]) is 3 sqrt(3) / 16. Question I.6 of
#1212.

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