Consider the function f(x)=x^1/x. It has maxima at x=e. and it is decreasing as we go along either direction away from x=e. The closest integers are 2,3. testing them would be sufficient. Which is greater 2^1/2 or 3^1/3? Simple powering ..L.C.M of 2 and 3 is 6. Powering them by 6 would give 8 and 9. So clearly 3^1/3 is larger.

The derivative of \(x^{\frac{1}{x}}\) is \(x^{\frac{1}{x}-2}(1- ln(x))\). Since this is undefined at 0, the derivative is zero when \(1=ln(x) \Rightarrow x=e\). So the only critical point is at \(e\) and it is easy to check it is a maximum value.

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestConsider the function f(x)=x^1/x. It has maxima at x=e. and it is decreasing as we go along either direction away from x=e. The closest integers are 2,3. testing them would be sufficient. Which is greater 2^1/2 or 3^1/3? Simple powering ..L.C.M of 2 and 3 is 6. Powering them by 6 would give 8 and 9. So clearly 3^1/3 is larger.

Log in to reply

The derivative of \(x^{\frac{1}{x}}\) is \(x^{\frac{1}{x}-2}(1- ln(x))\). Since this is undefined at 0, the derivative is zero when \(1=ln(x) \Rightarrow x=e\). So the only critical point is at \(e\) and it is easy to check it is a maximum value.

Log in to reply

3^(1/3) is the maximum.

Log in to reply