More fun with power towers, Part 2

Calculus Level 4

$\huge f(x)=x^{x^{x^{\cdot^{\cdot^\cdot}}}}$

Let $f(x)$ defined as the infinite power tower above, defined on the interval $[1,e^{1/e}]$ . Find the integer part of the derivative $f'(\sqrt{2})$ .

Compare with this .

The answer is 9.

2 solutions

Brian Charlesworth
Feb 13, 2016

This infinite tetration, or power tower , converges on the given interval and in fact achieves a maximum value of $e$ at $x = e^{\frac{1}{e}}$ , (see Dr. Bretscher's related question for a discussion on this matter). With confidence we can then write this equation as $y = x^{y} \Longrightarrow \ln(y) = y\ln(x)$ , which we can differentiate implicitly to find that

$\dfrac{y'}{y} = y'\ln(x) + \dfrac{y}{x} \Longrightarrow y'\left(\dfrac{1}{y} - \ln(x)\right) = \dfrac{y}{x} \Longrightarrow y' = \dfrac{y^{2}}{x(1 - y\ln(x))}$ .

Now $y = \sqrt{2}^{y}$ has, (on observation), solutions $y = 2$ and $y = 4$ , but as noted above the power tower achieves a maximum of only $e$ , thus $y(\sqrt{2}) = 2$ . Plugging $x = \sqrt{2}, y = 2$ into the equation for $y'$ yields that

$f'(\sqrt{2}) = \dfrac{2^{2}}{\sqrt{2}(1 - 2\ln(\sqrt{2}))} = \dfrac{4}{\sqrt{2}(1 - \ln(2))} = \dfrac{2\sqrt{2}}{1 - \ln(2)} = 9.2175367...$ ,

the integral part of which is $\boxed{9}$ .

Another clear and elegant solution! Thanks! (+1)

Question: How do we know that $f(x)$ is differentiable in the first place?

Otto Bretscher - 5 years, 4 months ago

I suppose we could note that the inverse function $f^{-1}(x) = \large x^{\frac{1}{x}}$ is continuous, increasing and one-to-one on $[1,e]$ , and thus the given function must have the same charactersitics on $[1,\large e^{\frac{1}{e}}]$ .

Brian Charlesworth - 5 years, 4 months ago

yes, that's how I thought of it: $g(x)=x^{1/x}$ is differentiable for $x>0$ and $g'(2)\neq 0$ . "Continuous, increasing and one-to-one" does not imply differentiability, of course.

Otto Bretscher - 5 years, 4 months ago

@Otto Bretscher – Yes, you're right. The function could have these attributes but also have corners and thus not be differentiable. (I think one could even construct such a function that is differentiable nowhere on the interval.) As you note, though, $g(x)$ is indeed differentiable for $x \gt 0$ , with $g'(x) = \large x^{\frac{1}{x} - 2}(1 - \ln(x))$ and $g'(2) = 0.108488... \ne 0$ .

Brian Charlesworth - 5 years, 4 months ago

@Brian Charlesworth – I don't think there is an increasing function that is nowhere differentiable... if my memory serves me right, an increasing function is differentiable almost everywhere.

Otto Bretscher - 5 years, 4 months ago

@Otto Bretscher – O.k., that's right, (again). I had the Cantor function in my head as a counterexample, but once I reviewed its characteristics I found that it was differentiable (with zero derivative) almost everywhere. (It's also not strictly increasing, and hence not one-to-one.)

Brian Charlesworth - 5 years, 4 months ago

Rishabh Jain
Feb 13, 2016

Are cobwebs required in this problem too ??.....:-)

Cobwebs are never required.... they are always optional. They are just a useful way to visualize results that can be obtained analytically.

Otto Bretscher - 5 years, 4 months ago

BTW how are you able to design such fresh and amazing problems...This question too was good and test's how a person can apply his equation solving as well as differentiation skills( Algebra + Calculus) .. I enjoyed this problem.. (+1) to the problem..

Rishabh Jain - 5 years, 4 months ago

These problems are not that fresh... my country man Euler was thinking about this stuff over 250 years ago. I noticed the error in that $x^{x^{x...}}=3$ problem... that made me think about those power towers.

Otto Bretscher - 5 years, 4 months ago

More fun with power towers, Part 2

The answer is 9.

2 solutions

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