Question

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

• A. $0$
• B. $\frac{1}{3}$
• C. $\frac{1}{2}$
• D. $\frac{1}{4}$

Answer 1

The correct option is D $\frac{1}{4}$

Let $y=x^{25}(1-x{)}^{75}$
$\Rightarrow \frac{dy}{dx}=25x^{24}(1-x{)}^{75}-75x^{25}(1-x{)}^{74}$
$=25x^{24}(1-x{)}^{74}(1-x-3x)$
$=25x^{24}(1-x{)}^{74}(1-4x)$
Clearly, critical point are $0,1/4$ and $1$.
Sign scheme of $\frac{dy}{dx}$
Thus, $x=1/4$ is the point of maxima.