Question:
On the interval \([0,1]\), the function \(f(x) = x^{25}(1-x)^{75}\) attains its maximum value at the point \(x =\)....
On the interval \([0,1]\), the function \(f(x) = x^{25}(1-x)^{75}\) attains its maximum value at the point \(x =\)....
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Differentiate, set to zero, and use the sign change to confirm a maximum.
Updated On: Oct 1, 2026
- \(0\)
- \(\frac{1}{4}\)
- \(\frac{1}{2}\)
- \(\frac{1}{3}\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
The function \(f(x) = x^{25}(1-x)^{75}\) is zero at both ends of \([0,1]\) and positive inside. So its maximum is at an interior critical point.
Step 2: Differentiate:
\[ f'(x) = 25x^{24}(1-x)^{75} - 75x^{25}(1-x)^{74} = 25x^{24}(1-x)^{74}\left[(1-x) - 3x\right] = 25x^{24}(1-x)^{74}(1 - 4x) \]
Step 3: Critical point:
On \((0,1)\), \(f'(x) = 0\) only when \(1 - 4x = 0\), that is \(x = \frac14\). For \(x < \frac14\) the derivative is positive and for \(x > \frac14\) it is negative, so \(x = \frac14\) is a maximum.
Step 4: Why the other options are wrong.
\(x = 0\) gives \(f = 0\), the minimum value. \(x = \frac12\) and \(x = \frac13\) lie in the region where \(f\) is already decreasing.
Final Answer:
The maximum is at \(x = \frac14\), option (B). \[ \boxed{x=\frac{1}{4}} \]
The function \(f(x) = x^{25}(1-x)^{75}\) is zero at both ends of \([0,1]\) and positive inside. So its maximum is at an interior critical point.
Step 2: Differentiate:
\[ f'(x) = 25x^{24}(1-x)^{75} - 75x^{25}(1-x)^{74} = 25x^{24}(1-x)^{74}\left[(1-x) - 3x\right] = 25x^{24}(1-x)^{74}(1 - 4x) \]
Step 3: Critical point:
On \((0,1)\), \(f'(x) = 0\) only when \(1 - 4x = 0\), that is \(x = \frac14\). For \(x < \frac14\) the derivative is positive and for \(x > \frac14\) it is negative, so \(x = \frac14\) is a maximum.
Step 4: Why the other options are wrong.
\(x = 0\) gives \(f = 0\), the minimum value. \(x = \frac12\) and \(x = \frac13\) lie in the region where \(f\) is already decreasing.
Final Answer:
The maximum is at \(x = \frac14\), option (B). \[ \boxed{x=\frac{1}{4}} \]
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