The maximum value of (1/x)^x is ______.

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The maximum value of (1/x)^x is underlinebb(e^(1/e)). Explanation: Let f(x) = (1/x)^x Taking log on both sides, we get log [f (x)] = x log 1/x $\Rightarrow$ log [f (x)] = x log x^-1 $\Rightarrow$ log [f (x)] = – [x log x] Differentiating both sides w.r.t. x, we get `1/("f"(x)) * "f'"(x) = - [x * 1/x + log x * 1] = - "f"(x) [1 + log x] $\Rightarrow$ f'(x) = - (1/x)^x [1 + log x]` For local maxima and local minima f'(x) = 0 -(1/x)^x [1 + log x] = 0 $\Rightarrow$ (1/x)^x [1 + log x]= 0 (1/x)^x $\neq$ 0 $\therefore$ 1 + log x = 0 $\Rightarrow$ log x = – 1 $\Rightarrow$ x = e–1 So, x = 1/"e" is the stationary point. Now f'(x) = -(1/x)^x [1 + log x] f"(x) = `-[(1/x)^x (1/x) + (1 + log x) * "d"/"dx" (x)^x] f"(x) = -[("e")^(1/"e") ("e") + (1 + log 1/"e") "d"/"dx" (1/"e")^(1/"e")] x = 1/"e" = -"e"^(1/"e") 1 < 0` maxima $\therefore$ Maximum value of the function at x = 1/"e" is `"f"(1/"e") = (1/(1/"e"))^(1/"e") = "e"^(1/"e")`

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