Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $n\ge 2$ be an integer and $a$ be a real number such that $|a|\le (n-1)n^{-\frac n{n-1}}$. Can we find the real solutions of the algebraic equation $$ x^n -x =a. $$

share|cite|improve this question

First, if $n$ is odd, then $f(x) = x^n-x$ is unbounded above and below, and so $f(x)=a$ has a solution for any $a$.

If $n$ is even, $f(x)$ is unbounded above and is convex, since $f''(x) = n(n-1)x^{n-2}\geq 0.$ Any local minimum of $f$ is therefore its global minimum.

Step 1: Find the global minimum of $f$ by solving $f'(x) =0.$

Step 2: What is the value of $f$ at this minimum?

Step 3: What can you conclude about the range of $f(x)?$

In neither case will you in general be able to solve for $x$ analytically.

share|cite|improve this answer
You can find closed-form solutions; they'll just be terribly complicated, and will involve special functions... – J. M. Jun 1 '13 at 5:37
Yes! I want a closed form. Thank you! – Chung. J Jun 1 '13 at 16:15

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.