given $\displaystyle f(x)=x^4+3x^2-x+1 $ show that there existat leastthree real numbers x such that $\displaystyle f(x)=3 $

how could I prove that there is at least three?

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- Dec 8th 2007, 05:53 PMakhayoonIntermedaite value theorm
given $\displaystyle f(x)=x^4+3x^2-x+1 $ show that there exist

**at least**three real numbers x such that $\displaystyle f(x)=3 $

how could I prove that there is at least three? - Dec 8th 2007, 06:00 PMJhevon
- Dec 8th 2007, 06:04 PMakhayoon
rechecked confirmed, this is how the quiz question was written.

- Dec 8th 2007, 06:14 PMJhevon
Below are the graphs of y = x^4 + 3x^2 - x + 1 and y = 3. as you can clearly see, they intersect only twice. namely at x = -0.6337532429 and x = 0.8738358889. thus there is something wrong with the question.

as you can see, the graph looks like a parabola, so, in fact, f(x) = c at most twice for any c - Dec 8th 2007, 06:17 PMtopsquark
- Dec 8th 2007, 06:17 PMakhayoon
ok, so would I use the I.V.T to prove them wrong?

- Dec 8th 2007, 06:22 PMJhevon
the question does not have a "prove or disprove" instruction, so i would not really recommend it. just tell your professor that there is something wrong with the question.

you can use IVT to prove there is at least**2**. to disprove that there are at least 3 cannot be done with IVT as far as i can see, but we can disprove it (if that's allowed) using Rolle's theorem - Dec 8th 2007, 09:27 PMcurvature