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How to use a returned pure function from say, Maximize

• To: mathgroup at smc.vnet.net
• Subject: [mg89258] How to use a returned pure function from say, Maximize
• From: christophsk at gmail.com
• Date: Sun, 1 Jun 2008 16:34:28 -0400 (EDT)

When I as Mathematica to maximize (over x) the expression

\frac{1}{2} \left(x \left(-2 \sqrt{1-x^2}+\sqrt{3-4 x (x+1)}+2\right)+
\sqrt{1-x^2}\right)

I am returned this (after simplifying) - you can try it yourself,

{1/2 Root[
47972 - 156976 #1 + 144464 #1^2 + 616 #1^3 - 55340 #1^4 +
18312 #1^5 + 1316 #1^6 - 964 #1^7 - 47 #1^8 + 64 #1^9 +
16 #1^10 &, 5], {x ->
Root[-351 + 1440 #1 + 2520 #1^2 - 12672 #1^3 - 8192 #1^4 +
38528 #1^5 + 18176 #1^6 - 44032 #1^7 - 19968 #1^8 +
16384 #1^9 + 8192 #1^10 &, 5]}}

Bear in mind, this is just an example, but it illustrates the point.
It is Ron Graham's Biggest Little Hexagon in case you want to look
it up.

I can obviously recover the numerical value. But if I actually wanted
to use this function, ie, substitute other expressions, manipulate it in some
way, how do I do this?

More to the point, I've always been a bit confused how to make use of
such a returned pure function.

Any help appreciated.