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Solve fails to identify the maximum of the entropy function

The entropy function has its maximum at x=1/2

Yet when one uses solve:

solve(derivative(-x*log(x) -(1-x)*log(1-x),x)==0,x)

sage returns something that is correct, but curiously non-specific

[log(x) == log(-x + 1)]

It almost seems like sage can't figure out that x=1/2 solves the above even after the assumption on x.