Solving x^4 + 3x + c = 0 with Mean Value & Rolle's Theorems

[powp]

Hello

I am really not getting this Mean Value Theorem and Rolle's theorem.

I have this question

Show how that the equation x^4 + 3x + c = 0 has at most two roots.

How do I know what interval to use for Mean Value Theorem and Rolles theorem?? Think this is the part that confusses me. I think that different results can be obtained if I pick different intervals.

Please help this lost student

Peter

 

[AKG]

Rolle's Theorem is, as you should be able to see, just a corollary of the Mean Value Theorem. Since you're concerned specifically with roots, it's probably best to just look at Rolle's Theorem. Now suppose it has more than two roots. You know that this would mean that it has at least three. Let's call them a, b, and c with a < b < c. So what intervals would Rolle's Theorem apply to? What would it tell you about the derivative of some point in these intervals? What would this tell you about the number of roots f' has? Compute f' and figure out how many roots it actually has. You'll come to a contradiction.

 

[HallsofIvy]

Suppose x^4 + 3x + c = 0 had 3 different roots: a< b< c. Apply Rolle's theorem to the intervals [a,b] and [b,c]. What does it tell you must be true about the derivative of x^4+ 3x+ c? What IS the derivative? Is it true?