This equation has only one root x=0. Proof that. Use derivatives for example

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- April 6th 2010, 02:33 AM #1

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- Apr 2010
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## e^x + x = 1

Hi everyone,

This is my first post in this forum so please forgive me if I haven't posted this question in the right place.

The problem is that I can't figure out how to solve for x in the following....

e^x + x = 1

The usual index and log rules don't seem to help much.

Any ideas?

Thanks.

- April 6th 2010, 03:02 AM #2

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- Mar 2010
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- April 6th 2010, 03:14 AM #3

- April 6th 2010, 03:23 AM #4
This is equivalent to the function

.

By inspection, it's easy to see that satisfies this equation.

Now, by Rolle's Theorem, if a function has more than one root, it has to turn around. So if we can show that the function does not have any turning points, then it can't have more than one root.

But since for all , this equation is not solvable (at least in the reals).

So there are not any turning points, which means there is only one root.

So is the only root.