TITLE: Maximum Value of the function $f(x)=x^n(1-x)^n$ for a natural number $n \geq1$ and $x\in[0,1]$.
QUESTION [5 upvotes]: I need some help with the following problem.
How, I can find the, Maximum Value of the function $f(x)=x^n(1-x)^n$ for a natural number $n \geq1$ and $x\in[0,1]$.

REPLY [4 votes]: $$f(x)=x^n(1-x)^n$$
$$f'(x)=n(x-x^2)^{n-1}(1-2x)=0$$
$$x-x^2=x(1-x)=0,x=0,x=1$$ and $f(0)=f(1)=0$
or 
$$1-2x=0,x=1/2$$ so maximum is for $x=1/2$
$$f(1/2)=(1/2)^{2n}$$