## James A Crawford

Having reached a sufficient station in life that I have either heard most of the typical interview questions that might come my own direction, or have the wit and nonsense to navigate my way through something that I don’t know and talk about the weather or something completely unrelated, this on-going collection of (technical) interview questions is intended just for fun. I encourage any readers to forward me their own most favorite technical questions at jk@am1.us. Don’t get up-tight though, after all, this is just for fun.

# Question #1:

What is the equivalent of j^{j} ?

This is very straight-forward to solve if we first write the base j value as j = exp( j π/2 ) based on Euler’s formula. Given that substitution, then

# Question #1B: Corollary to #1

What is the equivalent form of

This is the same as Question #1 except it has been re-cast in a slightly different form. The progression from this form back to Question #1 is given by

# Question #2: In honor of Al Thiele

Solve for (x,y) in the set of real pairs with x≠y given x^{y} = y^{x}

From this starting point, clearly

and

In order to have a solution with x≠y, the quantity z = x / ln( x ) must be multi-valued along the real line. A rough sketch of the solution space can be found by collecting a bit more information.

Specifically, the slope dz/dx must approach infinity as x+ nears unity because the log function in the denominator will blow up. Similarly, a second derivative computation shows that the slope dz/dx with large x becomes

It is also easy to show from this last equation that dz/dx= 0 for x= e.

Moving on to a complete solution to the problem, we can write

where c is an auxiliary variable. Solving this equation leads to

and substitution of this result back into y/x= c leads to

From this result, it is easy to further conclude that

The lesson here is to use simple graphical techniques to identify the behavior of a solution when possible. A picture is worth a thousand words.

The first part of these *dreaded interview questions* contains a total of 31 thought-provoking questions. To read further, just follow this link…