5

enter image description here

p.s I have no idea how to type math on this program so I just copied and pasted from a document.

p.p.s I also tried doing it with log base a and that was a travesty. This was my best attempt because it was the nicest looking

jd123
  • 349
  • 3
  • 11

2 Answers2

2

First observe that $x\le 0$ can't be a solution since $2a>2$ on the right side.

Then, use arithmetic and geometric mean for $xa^{1/x}$ and $a^x/x$. We obtain $$2a^{\frac{x+\frac1x}2}\le xa^{\frac1x}+\frac{a^x}x$$ Now use that $2\le x+\displaystyle\frac1x$ (again by A-G means) and that $a>1$ so we have $a=a^1\le \displaystyle a^{\frac{x+\frac1x}2}$, finally, $$2a\le 2a^{\frac{x+\frac1x}2}\le xa^{\frac1x}+\frac{a^x}x $$ and equality can hold only if $x=1$ by the last step.

Berci
  • 90,745
  • Great solution. I'm thinking I did a little hand-waving in mine, but I can't really see where. You did no hand-waving. :) – John Jan 07 '14 at 01:47
  • 2
    It smelled like an application of A-G inequalities, but couldn't clearly see how in the beginning, just started to apply it...:) – Berci Jan 07 '14 at 02:05
  • I've never used arithmetic-geometric inequalities but after googling it, I like this proof a lot! :) thanks! – jd123 Jan 07 '14 at 02:13
1

Multiplying by $x$ and rearranging we get:

$$x^2a^{1/x} -2ax + a^x = 0.$$

From the quadratic formula,

$$x = \frac{2a \pm \sqrt{4a^2 - 4a^{x + (1/x)}}}{2a^{1/x}}.$$

Now, $x$ must be positive, because $a$ is positive, and the square root is always less than $2a$.

However, the exponent under the square root, $x + \frac{1}{x}$ must be less than or equal to $2$ in order to keep the value under the square root positive.

This happens only at $x=1.$

John
  • 26,319
  • Very intuitive solution. I actually tried using the quadratic formula but gave up because of the x+1/x. Thanks! – jd123 Jan 07 '14 at 01:42
  • You're welcome. Also be sure to check out @Berci 's solution. I think actually it's more robust than mine, though I can't put my finger on why. – John Jan 07 '14 at 01:46