The function f(x) = x^{25}(1 - x)^{75} for x | vedclass

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Question

(D) Given $f(x) = x^{25}(1 - x)^{75}$.To find the maximum,we calculate the derivative $f'(x)$ using the product rule:$f'(x) = 25x^{24}(1 - x)^{75} + x

Vedclass · Accepted Answer

$1/4$

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(D) Given $f(x) = x^{25}(1 - x)^{75}$.To find the maximum,we calculate the derivative $f'(x)$ using the product rule:$f'(x) = 25x^{24}(1 - x)^{75} + x^{25} \cdot 75(1 - x)^{74} \cdot (-1)$$f'(x) = 25x^{24}(1 - x)^{74} [(1 - x) - 3x]$$f'(x) = 25x^{24}(1 - x)^{74} (1 - 4x)$Setting $f'(x) = 0$ for critical points in $(0, 1)$:$1 - 4x = 0 \implies x = 1/4$.Since $f(0) = 0$ and $f(1) = 0$,and $f(x) > 0$ for $x \in (0, 1)$,the function must attain its maximum at $x = 1/4$.

The function $f(x) = x^{25}(1 - x)^{75}$ for $x \in [0, 1]$ attains its maximum at $x = \dots$

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