Question

On the interval $[0,1]$, the maximum value of $x^{25}(1-x{)}^{75}$ is attained at $x=$

• A. 1
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. None ot these

Answer 1

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

Let $f\left(x\right)=x^{25}(1-x{)}^{75}$
$\Rightarrow f^{\prime }\left(x\right)=25x^{24}(1-x{)}^{75}-75x^{25}(1-x{)}^{74}$
$=25x^{24}(1-x{)}^{74}(1-4x)$
if $x<\frac{1}{4}$ $f^{\prime }\left(x\right)>0$,

while $x>\frac{1}{4}$ $f^{\prime }\left(x\right)<0$
f(x)is maximum at x=1/4.