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

Answer 1

Answer: (B)

Let $f\left(x\right) = x^{25}\left(1 - x\right)^{75}, x \in \left[0, 1\right]$
$\Rightarrow f '\left(x\right) = 25x^{24} \left(1 - x\right)^{75} - 75x^{25} \left(1 - x\right)^{74}$
$= 25x^{24} \left(1 - x\right)^{74} \left(1 - x\right) - 3x$
$= 25 x^{24} \left(1 - x\right)^{74} \left(1 - 4x\right)$

We can see that $f '\left(x\right)$ is positive for $x <\frac{1}{4}$
and $f '\left(x\right)$ is negative for $x > \frac{1}{4}$.
Hence, $f \left(x\right)$ attains maximum at $x = \frac{1}{4}.$