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=\frac m{m+n}`