At which point in the interval [0, 1] is the | vedclass

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(B) Let $f(x) = x^{25}(1 - x)^{75}$.To find the critical points,we differentiate $f(x)$ with respect to $x$:$f'(x) = 25x^{24}(1 - x)^{75} + x^{25} \cd

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$x = 1/4$

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(B) Let $f(x) = x^{25}(1 - x)^{75}$.To find the critical points,we differentiate $f(x)$ with respect to $x$:$f'(x) = 25x^{24}(1 - x)^{75} + x^{25} \cdot 75(1 - x)^{74}(-1)$$f'(x) = 25x^{24}(1 - x)^{74} [(1 - x) - 3x]$$f'(x) = 25x^{24}(1 - x)^{74} (1 - 4x)$.Setting $f'(x) = 0$,we get critical points at $x = 0$,$x = 1$,and $x = 1/4$.Evaluating the function at the critical points and endpoints:$f(0) = 0^{25}(1 - 0)^{75} = 0$$f(1) = 1^{25}(1 - 1)^{75} = 0$$f(1/4) = (1/4)^{25}(1 - 1/4)^{75} = (1/4)^{25}(3/4)^{75} > 0$.Since $f(x) \geq 0$ for all $x \in [0, 1]$ and the value at $x = 1/4$ is positive,the function attains its maximum at $x = 1/4$.

At which point in the interval $[0, 1]$ is the function $f(x) = x^{25}(1 - x)^{75}$ maximum?

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