In the interval [0, 1] the function $x^{25}{(1-x)}^{75}$takes a maximum value at

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^{25}{\left(1-\mathrm{x}\right)}^{75} \\ \mathrm{logf}\left(\mathrm{x}\right)=25\mathrm{logx}+75\log \left(1-\mathrm{x}\right) \\ \frac{{\mathrm{f}}^{\mathrm{I}}\left(\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{x}\right)}=\frac{25}{x}+\frac{75}{1-x}\left(-1\right) \end{gathered}$$

$\mathrm{f}\left(\mathrm{x}\right)$ is min or max

$$\begin{gathered} \Rightarrow {\mathrm{f}}^{\mathrm{I}}\left(\mathrm{x}\right)=0 \\ \frac{25}{x}=\frac{75}{1-\mathrm{x}} \\ 1-\mathrm{x}=3\mathrm{x}\Rightarrow 1=4\mathrm{x} \\ \Rightarrow \mathrm{x}=1/4 \end{gathered}$$

$\mathrm{f}\left(\mathrm{x}\right)$ is max at x=1/4