Number of distinct real roots with $e^{-x}$ in the equation

Question

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$

I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but here what about $e^{-x}$

Answer 1

Let $$f(x)=13x^{13}-e^{-x}-1$$ then $$f'(x)=169x^{12}+e^{-x}$$ which is positive. Therefore f is strictly ascending. The root of the equation is the point of intersection of the graphs of the functions $$g(x)=13x^{13}$$ and $$h(x)=e^{-x}-1$$

Or observe that $f(0)<0$ and $f(1)>0$, since $f$ is continuous, $f(x)=0$ has a root in $(0,1)$.

Answer 2

First of all: $f(x) = 13x^{13}-e^{-x}-1$ has at most one real root, because $f'(x) = 169x^{12}+e^{-x} > 0$. It has one real root $0<x_0< 1$by inspection.

Answer 3

If you know about the derivative, this can be done directly. Let $f(x) = 13 x^{13} - e^{-x} - 1$; then

$$f'(x) = 169 x^{12} + e^{-x}$$

which is always positive; hence $f$ is strictly increasing, and there is at most $1$ real solution. Finally, since $f(0) < 0$ and $f(1) > 0$, the real solution is between $0$ and $1$ (Wolfram gives about $.844$).

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Some comments on not using calculus: One can notice that (just since $0 < e^{-x} < 1$ for positive $x$) we have $f(x) > 0$ for all $x > 1$; in fact, we can get the stronger result that $f$ has no zeros whenever $13 x^{13} > 2$. Solving for $x$, this means there are no solutions on $(\sqrt[13]{2/13}, \infty)$, or about $(.866, \infty)$ (pretty good!).

Likewise, and for the same reasons, there are no solutions when

$$13 x^{13} < 1 + e^{-1}$$

or $(-\infty, .841)$ (also pretty decent!).

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I don't immediately see a way to prove uniqueness (or even existence) without some basic notions from calculus.

Answer 4

To 'find' the number of distinct real roots:

You need to think in terms of its graphs. Consider the general shape of the graph of $1+e^{-x}$ and $13x^3$ and guess the number of times the graph will cut each other.

To 'prove' your answer:

Show that the function (the L.H.S) is monotone increasing (differentiation?). Also find two points where the function evaluates positive and negative and use intermediate value theorem to claim the existence of a root between the two values.