Find the maximum of the function $$f(x)=x^{1/x}$$ and the value of $x$ which gives the maximum value?
Find the maximum of $f(x)=x^{1/x}$
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$\begingroup$
calculus
optimization
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2Like every other maximum problem: differentiate, set the derivative to 0, solve for $x$, check whether any of the solutions happen to be maxima. – 2012-03-03
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3It's a bit easier if you consider $\ln f(x)$ instead. – 2012-03-03
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0@HaraldHanche-Olsen: How does taking the logarithm make it easier? – 2012-07-09
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0@celtschk: Just take a look at J.D.'s answer. – 2012-07-10
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2I'm looking. But I still don't see how it's easier than just deriving $x^{1/x}$ directly: $(x^{1/x})' = (1/x)x^{1/x-1} + x^{1/x}\ln x \cdot (-1/x^2) = x^{1/x-2}(1-\ln x)$ – 2012-07-10
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0@HaraldHanche-Olsen: Sorry, forgot the at-addressing. – 2012-07-10
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0@celtschk: No, if you know the rule you just used, it's not a great improvement. But unless you have learned a specific rule for differentiationg $f(x)^{g(x)}$, this relies on a modest bit of multivariable calculus, using the partial derivatives of $x^y$ and the multivariable chain rule. But it's not really a big deal (which is why I put the words “a bit” in my comment). – 2012-07-11
2 Answers
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Let $y = x^{1/x}.$ So $$ \ln y = (1/x)\ln x. $$ Differentiate both sides w.r.t $x$, we get $$ y'/y = (1/x)(1/x) + (-1/x^2) \ln x. $$ Rearranging, we have $$\dfrac{d}{dx}(x^{1/x}) = (1-\ln x)x^{1/x - 2} $$ Set $\dfrac{d}{dx}(x^{1/x}) = 0$ and work from there to get the maximum.
$$\text{(Hint: maximum occurs at x = e.)}$$
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0What is y'/y? i don't get it – 2012-03-03
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2I differentiated both sides of the equation $\ln y = (1/x) \ln x$. $\dfrac{d}{dx} \ln y = (\dfrac{d}{dx} y)/y$ by the rules of derivatives of logarithmic functions, and $y'$ is a shorthand for $\dfrac{d}{dx} y$. – 2012-03-03
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This is not original with me.
If we know that $e^x \ge 1+x$ with equality only when $x = 0$, $e^{(x-e)/e} \ge 1 + (x-e)/e = x/e$ or $e^{x/e} \ge x$ or $e^{1/e} \ge x^{1/x}$ with equality only if $x = e$.
Ta-dah!
At no time do the fingers leave the hands!
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1+1 for presentation! I watch them magic tricks close though, I do. – 2012-07-09
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0I like this! It is so smooth just like exp itself. – 2018-04-09