Finding coordinates of maximum with unknown powers $f(x)=(x^m)*((1-x)^n)$ on a line segment [0;1]

Question

So I need to find the coordinates of the maximum of a function on a line segment.So I approached this by taking the derivative and equated it with $0$. But due to $m$ and $n$ which are unknown, how do I find coordinates of the maximum?

Answer 1

I suppose $m,n$ are positive integers.

$f'(x)=m x^{m-1} (1-x)^n-n x^m (1-x)^{n-1}=x^{m-1} (1-x)^{n-1} [m (1-x)-n x]$

$$f'(x)=0\to x=0;\;x=1;\;m (1-x)-n x=0$$

$$m-mx-nx=0;\;(m+n)x=m\to x=\frac{m}{m+n}$$

$f'(x)=0\to x_1=0;\;x_2=1;\;x_3=\frac{m}{m+n}$

$$f''(x)=x^{m-2} (1-x)^{n-2} \left(m^2 (x-1)^2+m (x-1) ((2 n-1) x+1)+(n-1) n x^2\right)$$ Thus we are not sure if $x_1$ and $x_2$ are minima, maxima or inflection points. It depends on the actual values of $m$ and $n$.

$x_3\in[0,1]$ because $0\le \frac{m}{m+n}\le 1$. If this happens, then $x_3$ is a point of relative maximum for $f(x)$ because $$f''(x_3)=-\frac{m^2 \left(\frac{m}{m+n}\right)^{m-3} \left(\frac{n}{m+n}\right)^n}{n}<0$$