MaplePrimes - answers and comments on Question, Can't maple even maximize such a simple function?

a cleaner copy of the commands

dinonykus — Tue, 24 Oct 2006 05:40:08 Z

> maximize(t^2*(1-t)^2*(1+t^2)*(1+(1-t)^2), t = 0 .. 1); > maximize((1/2-s)^2*(1/2+s)^2*(1+(1/2-s)^2)*(1+(1/2+s)^2), s = (-1)/2 .. 1/2); > maximize((1/4-s^2)^2*((5/4+s^2)^2-s^2), s = (-1)/2 .. 1/2); > maximize((1/4-u)^2*((5/4+u)^2-u), u = 0 .. 1/4); The first three lines failed to give the maximum. The last line gave the maximum 25/256. The original function t^2*(1-t)^2*(1+t^2)*(1+(1-t)^2) simplifies to (1/4-u)^2*((5/4+u)^2-u) by setting u=(1/2-t)^2. My question is: Is there a simpler way to directly give the max without going through the pain of simplification/transformation? Thanks a lot.

thanks

dinonykus — Tue, 24 Oct 2006 06:06:01 Z

Yes, I have maple 10, which actually gave a better result:) > Optimization:-Maximize(t^2*(1-t)^2*(1+t^2)*(1+(1-t)^2), t = 0 .. 1); [0.0976562500000000000, [t = 0.500000000000000000]] But, is there a way to give the exact answer rather than a numerical one? I am thinking about a possiblity that the max value might be sqrt(5)/16, attained at t=sqrt(5)/2. It would be preferable to have an exact answer in such cases. Thanks a lot.

Optimization and a workaround

alec — Tue, 24 Oct 2006 06:07:56 Z

A simple workaround is to add a decimal dot to one of the numbers in your function. For example,

f:=t^2*(1.-t)^2*(1+t^2)*(1+(1-t)^2):

maximize(f,t=0..1,location);

         0.09765625000, {[{t = 0.5000000000}, 0.09765625000]}

Also, if you have relatively new version of Maple that has the Optimization package, you could find the maximum as

Optimization:-Maximize(t^2*(1-t)^2*(1+t^2)*(1+(1-t)^2), t = 0 .. 1);

      [0.0976562500000000002, [t = 0.499999999999999888]]

I see

dinonykus — Tue, 24 Oct 2006 06:20:51 Z

That's a clever way to go about. So it seems Maple can't directly give exact answers? Thanks a lot anyway. I appreciate your help.

Another workaround

alec — Tue, 24 Oct 2006 06:24:12 Z

Actually, there is another workaround,

_EnvExplicit:=false:
maximize(t^2*(1-t)^2*(1+t^2)*(1+(1-t)^2),t=0..1,location);

                       25                25
                       ---, {[{t = 1/2}, ---]}
                       256               256

Yay, hooray

dinonykus — Tue, 24 Oct 2006 06:28:07 Z

I'll remember this one. Thank you so much.