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### Maximizing Output of a Restricted Function

```
Date: 11/1/96 at 23:37:55
From: Geoffrey Lovelace
Subject: Maximizing Functions

A teacher offers extra credit to a math student with the following
condition: the number of extra credit points will be decided with a
function. The student writes any function f(x) and turns it in to the
teacher. The teacher than chooses any complex number in the domain of
f(x). The output will be the number of points added to the student's
score if positive and the number subtracted if negative. If the output
has an imaginary part, the real part is subtracted and an additional
10^6 points are deducted from the student's score (a bad deal).

The function, when simplified (before an input is entered), may not
have a constant term, may not be a constant function (where the
variables divide out to one, for instance), and may have no number
(coefficient, exponent, etc) greater than 3. What function should the
student write to ensure the maximum number of points?

At first, I tried using a rational function with an absolute value of
x * the absolute value of -3 cubed over the absolute value of x (to
restrict x). However, then I realized that if the teacher chooses x
such that -1<x<1, the number of points cubed on the numerator became
pathetically small (less than 1E500 and beyond).

Next, I realized that this approach wouldn't work at all, since an
input like 3+2i would yield an imaginary number. I tried using an x!
but was unsure whether this would restrict the domain successfullly.
Things like x squared over x squared break the 'no constant function'
rule, as does x^3^3^3... (that simplifies).

I have not yet, then, found a solution that satisfies this problem
with a domain that includes complex numbers (I'm not even sure about
one that satisfies real numbers). I cannot even determine if there is
an answer that works! On the other hand, this problem seems to have
many potential answers (I just cannot discover any that satisfy every
condition.)
```

```
Date: 11/05/96 at 21:06:17
From: Doctor Rob
Subject: Re: Maximizing Functions

I don't know the best function to use, either, but I did find some
very good ones.

To reduce the problem with complex inputs to one with nonnegative real
ones, you can replace every occurrence of x with |x|.  Now things are
simpler.  How about something like 3^(|x|+3), which will give the
student at least 27 points?  Or 3^[3^(|x|+3)], which will give the
student at least 3^27 points?

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Functions
High School Imaginary/Complex Numbers
High School Puzzles

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