Question

On the interval $[0,1]$ the function $x^{125}{(1-x)}^{75}$ takes the maximum value at the points

• A. $\frac{3}{8}$
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. $\frac{5}{8}$

Answer 1

The correct option is D $\frac{5}{8}$

Let $f\left(x\right)=x^{125}(1-x{)}^{75}$
$f^{\prime }\left(x\right)=125x^{124}(1-x{)}^{75}-75x^{125}(1-x{)}^{74}=x^{124}(1-x{)}^{74}(125(x-1)-75x)$
For maximum value of $f\left(x\right)$
$f^{\prime }\left(x\right)=0=x^{124}(1-x{)}^{74}(125(x-1)-75x)\Rightarrow x=\frac{5}{8}$ $atx=(0,1)$ $f^{\prime \prime }\left(x\right)$ will vanishes.