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April 15th, 2013, 09:41 PM   #1
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Intermediate Value Theorem

Hi all

Strange question here involving the intermediate value theorem.

Consider the equation x^3 + x +e^x = 0

Use the intermediate value theorem to show there is a solution on the interval (-1,0) and show that there is exactly one real solution of the equation.


I've been trying at this one for a couple of hours now and I cant seem to make good headway. My equation fall apart before too long. I have an idea on solving the second half of the question, showing there is exactly on real solution, and that is by graphing it and then possibly trying to manipulate it?


Any help is greatly appreciated. Thank you!
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April 15th, 2013, 10:02 PM   #2
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Re: Intermediate Value Theorem

It is not as difficult as it may seem, let







So... the y value went from negative to positive... it must have hit zero at least once... that's the intermediate value theorem.

To show it hit zero only once take the derivative,



So... f'(x) is positive for all real x (do you see why?), this means the function is always increasing... therefore it cannot cross the x axis more than once.

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April 16th, 2013, 12:40 AM   #3
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Re: Intermediate Value Theorem

Oh my. I just had the biggest facepalm moment.

Thank you so much, you just saved me a sleepless night
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