Question 977299
x+y=1
y=1-x
-
{{{x(1-x)^4+(1-x)x^4}}}
{{{x(1-x)((1-x)^3+x^3)}}}
{{{x(1-x)(-x^3-x^2-3x+1+x^2)}}}
{{{x(1-x)(-x^2-3x+1)}}}
{{{(-1)x(1-x)(x^2+3x-1)}}}


You may be able to identify roots of the factored expression and on which intervals is the expression positive.  You could check using iterations of numerical values to find max values as near as you want for any maximum values which exist.


The ROOTS  (not any extremes)  are  0, 1, (-3-sqrt(13))/2, (-3+sqrt(13))/2.
Remember you wanted positive values for x, and that {{{0<x<1}}}.


I have not worked through to any finished answer; just planned for the method.