if m,n equal to or greater than 2 are integers, find the critical points of f(x) x^m (1-x)^n

there are 3 critical points what are they? HELP

Question

if m,n equal to or greater than 2 are integers, find the critical points of f(x) x^m (1-x)^n

there are 3 critical points what are they? HELP

anonymous · Answer

$$f(x)=x^m(1-x)^n~~\Rightarrow~~f'(x)=mx^{m-1}(1-x)^n-nx^m(1-x)^{n-1}$$
When is this equal to zero?
$$\begin{align*}mx^{m-1}(1-x)^n-nx^m(1-x)^{n-1}&=0\
x^m(1-x)^n\left(\frac{m}{x}-\frac{n}{1-x}\right)&=0
\end{align*}$$
Two of the solutions are readily apparent, $x=0$ and $x=1$.
$$\begin{align*}\frac{m}{x}-\frac{n}{1-x}&=0\
\frac{m(1-x)-nx}{x(1-x)}&=0&\text{Presumably, }x\text{ is neither 0 nor 1.}\
m(1-x)-nx&=0\
m-(m+n)x&=0\
m&=(m+n)x\
x&=\frac{m}{m+n}
\end{align*}$$

anonymous · Answer

See the graph for m=2 and n=3 http://www.wolframalpha.com/input/?i=plot++%281+-+x%29%5E3+x%5E2+from+x%3D-.1+to+x%3D1.4 You cansee the 3 critical points

anonymous · Answer

thank you