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Application of Rolle's Theorem

Just a quick question.

I have a function, say f.

I was asked if it had a solution on an interval (which it does, by change of sign and continuity).

I then needed to decide whether the solution is unique. I applied Rolle's theorem and found that the derivative was always greater than or equal to zero on the interval - but I'm not sure what my conclusion should be?!

I know if it is strictly greater than zero the solution is unique, but its not quite...?
Original post by ninegrandstudent
Just a quick question.

I have a function, say f.

I was asked if it had a solution on an interval (which it does, by change of sign and continuity).

I then needed to decide whether the solution is unique. I applied Rolle's theorem and found that the derivative was always greater than or equal to zero on the interval - but I'm not sure what my conclusion should be?!

I know if it is strictly greater than zero the solution is unique, but its not quite...?

Which function are you working with here?

Generally, if you contradict Rolle's after assuming there are more roots than you expect, then you're done. If this fails, i.e. you find a point where the derivative is 0 in between a given pair of roots, then you need to search for the existence of another root (just as you did before using IVT/cty/change of sign kinda stuff) and then try to contradict Rolle's on the new intervals.

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