Question

On the interval \([0,1]\), the function \(f(x) = x^{25}(1-x)^{75}\) attains its maximum value at the point \(x =\)....

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

Hint:

Differentiate, set to zero, and use the sign change to confirm a maximum.

Answer 1

The Correct Option is B

Step 1: Shortcut for x^m (1-x)^n:
Maximise $\ln f = 25\ln x + 75\ln(1-x)$.

Step 2: Differentiate:
$\frac{25}{x} - \frac{75}{1-x} = 0$, so $25(1-x) = 75x$, giving $25 = 100x$ and $x = \frac14$.

Step 3: General rule:
The maximum of $x^m(1-x)^n$ is at $x = \frac{m}{m+n} = \frac{25}{100}$. The second derivative of $\ln f$ is negative, confirming a maximum.

Final Answer:
The maximum occurs at x = 1/4, option (B). \[ \boxed{x=\frac{1}{4}} \]