Rolle's Theorem question regarding # of real roots

[WatkinsonMolly]

Sorry, I don't mean to ask 2 questions in one day, but the book examples on Rolle's theorem are really, really bad - so bad that both examples end up having no solutions. The equation is P(x) = x^11 - 2x^2 -5. I've calculated P'(x) = 11x^10 -4x . How do I actually apply Rolle's theorem to find the number of real roots it has ( or has at most).

 

[galactus]

Check f(0), f(1), and f(2).

$\displaystyle f(0)=-5$

$\displaystyle f(1)=-6$

$\displaystyle f(2)=2035$

See how it changes from negative to positive between x=1 and x=2?.

 

[WatkinsonMolly]

Yes, I see that now, thanks. I should have specified, however, that this question was multiple choice with answers as follows -

A) P has 11 real roots

B) P has at most 3 real roots

C) P has no real roots

D) P has at most 2 real roots

E) P has 10 real roots

Obviously, A, C, and E are thrown out just by graphing it, but I don't really see how you can just look at the equation without graphing it and apply Rolle's Theorem to see that either B or D are the correct answer. When you look at end behavior, the values will get really huge as X gets larger, and much smaller as x gets smaller, but how do you apply the approximation? How can I tell that P has at most 2 or 3 real roots? This book is really irritating me with its explanation, so thanks for helping me clear it up

 

[lookagain]

Yes, I see that now, thanks. I should have specified, however, that this question was multiple choice with answers as follows -

A) P has 11 real roots

B) P has at most 3 real roots $\displaystyle \text{Imaginary roots come in even numbers.}$
$\displaystyle \text{So potentially there could be an odd number of roots, }$
$\displaystyle \text{such as 3 real roots.}$

C) P has no real roots

D) P has at most 2 real roots$\displaystyle \text{Because the imaginary roots (when they occur)}$
$\displaystyle \text{ come in groups with an even number of roots, then there could }$
$\displaystyle \text{not be a polynomial with 2 real roots.}$

E) P has 10 real roots

Obviously, A, C, and E are thrown out just by graphing it, but I don't really see how you can just look at the equation without graphing it and apply Rolle's Theorem to see that either B or D are the correct answer. When you look at end behavior, the values will get really huge as X gets larger, and much smaller as x gets smaller, but how do you apply the approximation? How can I tell that P has at most 2 or 3 real roots? This book is really irritating me with its explanation, so thanks for helping me clear it up

.