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I've been working on some problems and this particular one caught me off guard. Just by reading it, I thought it was a very simple problem where I equate the two equations and solve for $x$. However, my attempt was futile and I am stuck.

Prove that the graphs of the functions $y = x \sin x$ and $y = x^3 + x + 1$ intersect.

Hint: Check the values at x = -1, and x = 0

Any hints or help are greatly appreciated! Thank you for your time.

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To find points of intersection, equate the two equations and solve for $x$. – Mhenni Benghorbal Dec 8 '13 at 3:56
@Mhenni, I would like to see you solve $x\sin x=x^3+x+1$ for $x$. – Gerry Myerson Dec 8 '13 at 3:57
Consider $f(x) = x \sin x - (x^3+x+1)$ and apply the hint. – Macavity Dec 8 '13 at 3:58
@GerryMyerson: I leave it to you you to work it out :). – Mhenni Benghorbal Dec 8 '13 at 3:59
up vote 8 down vote accepted

See if you can apply the Intermediate Value Theorem. If a continuous function changes sign on interval, it must be zero somewhere on the interval.

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(+1) nice answer. – Mhenni Benghorbal Dec 8 '13 at 3:59
Thanks for the hint! I got it now! Thank you all for your time :) – wonggr Dec 8 '13 at 4:19

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