Woe is me. Up is down, left is right, and apparently .
Here's the problem from my old textbook:
Here's the very (un)helpful little hint they give:Discuss the following false "proof":
Suppose is a solution to . Clearly cannot be 0; hence .
Use this expression to substitute for 1 in the first conditional equation:
We now infer that .
Substituting this value into our original equation, we have
So, my questions are, in order of importance:
0. How do I make a cube root symbol?
1. What kind of sick person dreams up false proofs?
2. How is the hint supposed to help? If I transform the step to , and from there to , I now have a more complicated expression -- one for which the solution is still -1 or +1, neither of which still play well in .
3. If I encountered this kind of problem without this helpful hint, how would I know to use it? It seems like a singular rabbit to pull out of a hat...
4. This is the big one... Following normal algebraic processes, starting from a reasonable assumption, this horrific result, uh, resulted. Is this a failure of algebra, or of the idiot user attempting to make it work? Is there a tell-tale sign anywhere along the steps that something went wrong?
I vaguely remember this problem from as far back as high school, and I think it even made sense at the time -- but that was a rather long time ago from my perspective. May it please the court, I humbly beseech any assistance anyone can muster...
Attempting to Relearn Calculus
For Masochism and Glory