How is the graph of $\large x^{\frac{1}{x}}$ going to look like?

Calculus Level 3

$\large 1, ~ 2^{\frac{1}{2}}, ~ 3^{\frac{1}{3}}, ~ 4^{\frac{1}{4}}, ~ 5^{\frac{1}{5}}, ~ 6^{\frac{1}{6}}, \cdots$

Find the largest surd in the above series. Give your answer correct to three decimal places.

The answer is 1.442.

1 solution

Chew-Seong Cheong
Sep 20, 2016

Let $y = x^\frac 1x$ . Then $\dfrac {dy}{dx} = x^\dfrac 1x \left(\dfrac {1-\ln x}{x^2} \right)$ . And $\dfrac {dy}{dx} = 0$ $\implies \ln x = 1$ $\implies x = e$ . We note that $\dfrac {d^2y}{dx^2}\bigg|_{x=e} < 0$ , therefore, the maximum $y$ , $y_{max} = e^\frac 1e$ for all real $x$ . The nearest term in the series to $e^\frac 1e$ is $3^\frac 13 \approx \boxed{1.442}$ .

I should've marked it as a calculus problem but in a bit of rush I accidentally chose Number Theory for this.

Anyways, a straight forward approach as always sir.

Tapas Mazumdar - 4 years, 8 months ago

It is okay to mark it as Number Theory. It is about numbers. It is just that the method to solve it is Calculus.

Chew-Seong Cheong - 4 years, 8 months ago

Tapas, don't use Huge. It makes the problem looks childish. Punctuation rules do not need a space before comma ",". There should be a space after comma and LaTex takes care of that. No need to assume that 1 is a surd because it does not affect the answer and it is not necessary that all the elements of the series must be a surd.

Chew-Seong Cheong - 4 years, 8 months ago

Okay, I'll take care of this from now. Thank you sir. :)

Tapas Mazumdar - 4 years, 8 months ago

How is the graph of x 1 x \large x^{\frac{1}{x}} x x 1 ​ going to look like?

The answer is 1.442.

1 solution

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How is the graph of $\large x^{\frac{1}{x}}$ going to look like?