Mathematics - Application of Derivative Question with Solution | TestHub

Let $\mathrm{f}(\mathrm{x})={\mathrm{x}}^{25}(1-\mathrm{x}{)}^{75},\mathrm{x}\in [0,1]$ $\begin{array}{rl}\Rightarrow f^{\prime }(x)& =25x^{24}(1-x{)}^{75}-75x^{25}(1-x{)}^{74}\\ & =25x^{24}(1-x{)}^{74}\{(1-x)-3x\}\\ & =25x^{24}(1-x{)}^{74}(1-4x)\end{array}$
We can see that $f^{\prime }(x)$ is positive for $\mathrm{x}<\frac{1}{4}$ and $f^{\prime }(x)$ is negative for $x>\frac{1}{4}$. Hence, $f(x)$ attains maximum at $\mathrm{x}=\frac{1}{4}$.