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

Question

Aicharya · Accepted Answer

$\begin{aligned} & f(x)=x^{25}(1-x)^{75} \ & \Rightarrow f^{\prime}(x)=25 x^{24}(1-x)^{75}-x^{25} \cdot 75(1-x)^{74} \ & =25 \cdot x^{24} \cdot(1-x)^{74}(1-4 x)\end{aligned}$ $\begin{aligned} & \Rightarrow f(x) 	ext { takes maximum value at } x=\frac{1}{4} \ & \end{aligned}$

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

Solution