Question

If m,n≥2 are integers, find the critical points of f(x)=xm(1−x)n. The field below accept a list of numbers or formulas separated by semicolons (e.g. 2; 4; 6 or x+1; x−1.) The order of the list does not matter. Enclose numerators and denominators in parentheses. For example, (a−b)/(1+n).

Answer 1

By the product rule,

$f(x)=x^m(1-x)^n$

has derivative

$f'(x)=mx^{m-1}(1-x)^n-nx^m(1-x)^{n-1}=x^{m-1}(1-x)^{n-1}(m(1-x)-nx)$

$\implies f'(x)=x^{m-1}(1-x)^{n-1}(m-(m+n)x)$

Critical points occur where the derivative vanishes:

$x^{m-1}(1-x)^{n-1}(m-(m+n)x)=0$

$x^{m-1}=0\text{ or }(1-x)^{n-1}=0\text{ or }m-(m+n)x=0$

$\implies x=0\text{ or }x=1\text{ or }x=\dfrac m{m+n}$