I am reading a paper and they mention a hump function maximization: I am trying to prove the point of maximization:

$m=(1-x{)}^{1-\sigma}{x}^{\sigma}$ where $x,\sigma \in [0,1]$

It is said that m is a hump-shaped function of x maximized at $x=\sigma $, where $\sigma $ is a parameter that is assumed to be fixed here.

My attempt:

First derivative with respect to x: $(1-x{)}^{-\sigma}{x}^{\sigma}+(1-x{)}^{1-\sigma}{x}^{\sigma -1}=0$

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

Second derivative with respect to x: $-(1-x{)}^{-\sigma -1}{x}^{\sigma}+(1-x{)}^{-\sigma}{x}^{\sigma -1}-(1-x{)}^{-\sigma}{x}^{\sigma -1}+(1-x{)}^{1-\sigma}{x}^{\sigma -2}$

I couldn't get to the given result based on the above. Any clarification would be appreciated!

$m=(1-x{)}^{1-\sigma}{x}^{\sigma}$ where $x,\sigma \in [0,1]$

It is said that m is a hump-shaped function of x maximized at $x=\sigma $, where $\sigma $ is a parameter that is assumed to be fixed here.

My attempt:

First derivative with respect to x: $(1-x{)}^{-\sigma}{x}^{\sigma}+(1-x{)}^{1-\sigma}{x}^{\sigma -1}=0$

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

Second derivative with respect to x: $-(1-x{)}^{-\sigma -1}{x}^{\sigma}+(1-x{)}^{-\sigma}{x}^{\sigma -1}-(1-x{)}^{-\sigma}{x}^{\sigma -1}+(1-x{)}^{1-\sigma}{x}^{\sigma -2}$

I couldn't get to the given result based on the above. Any clarification would be appreciated!