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The function $f_n(x)$ is defined by the equation $f_n(x) = x^{n–x}$, where $0 \le x \le n$ and $x$,$n$ are integers. What is the smallest value of $n$ for which the maximum of $f_n(x)$ occurs when $x = 3$ ?

$f_1(0)=0$

$f_1(1)=1^{1-1}=1$

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The question has been revised. –  Rajesh K Singh Jan 25 '13 at 7:14
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2 Answers

Hint: Use the fact that $x^{n-x}=e^{(n-x)(\ln x)}$ and use the ordinary procedure for maxima/minima.

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Take first and second derivatives of $f_n(x)$. If you want the maximum when $x=3$, make sure that $f'_n(3)=0$ and $f''_n(3)\leq 0$.

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