Roots of a polynomial and differenciaton

[babita]

Homework Statement

I read that if f'(x) is zero once in [a b] then f(x) has maximum two real roots.
Why maximum? Shouldn't it be exactly 2?
Or it has something to do with the case of repeated roots?

Homework Equations

The Attempt at a Solution

was thinking as in figure

 

[mtayab1994]

Well in your case it has exactly two real roots, but in another case you might only have one real root or none. Try it out and draw a graph and see what you get. Often there would be one real root and one imaginary root.

 

[babita]

oh...like a quadritic equation with D=o , will have f'(x)=0 at -b/2a but it does not have any real roots ?

 

[babita]

Thanks :)

 

[mtayab1994]

Ok take for example x^2+x+1. Does this have any real roots? If yes then what are they?

 

[babita]

no it doesn't have any real roots. but f'(x)=0 at x=-0.5
right?

 

[babita]

sry typing mistake i there meant D<0

 

[mtayab1994]

Yes you are correct. You take it's derivative, and you have to see where that derivative equals 0. Like in the equation i gave you 2x+1=0 implies that 2x+1=0 if and only if x=-0.5.

 

[mtayab1994]

sry typing mistake i there meant D<0

Yes if you have the a negative discriminant, then you have two complex roots which are complex conjugates of one another.