Let $f\left(x\right)=x^{25}{\left(1-x\right)}^{75}\forall x\in \left[0,1\right]$ , then $f\left(x\right)$ assumes its maximum value at

$f\text{'}\left(x\right)=x^{25}\times 75{\left(1-x\right)}^{74}\left(-1\right)+{\left(1-x\right)}^{75}\times 25x^{24}$

             $=25x^{24}{\left(1-x\right)}^{74}\left\{-3x+\left(1-x\right)\right\}\forall x\in R$

$f\text{'}\left(x\right)=0\Rightarrow x=\frac{1}{4}\in \left(0,1\right)$

Note that$f\left(0\right)=f\left(1\right)=0$ and

$f\left(\frac{1}{4}\right)=\frac{3^{75}}{4^{100}}$ . So$f\left(x\right)$ is maximum at$x=\frac{1}{4}$ .