Sorry, I have hardly been able to be at the site for various reasons, but let me not be like certain of our students who never come back.@epenguin I think you have the right idea, but it's a bit hard for me to follow what you've written.
If [itex]p(x)=x^3[/itex], then [itex]p'(0)=p''(0)=0[/itex] and [itex]0[/itex] is a turning point according to your definition.
I don't understand this. If [itex]p(x)=x^4-1[/itex], then the point [itex]0[/itex] is 'such a point' ([itex]p'(0)=p''(0)=0[/itex]), but the only zero of [itex]p'[/itex] is zero. Do you mean to say that there are at least two roots of [itex]p'[/itex] counted with multiplicity (because the point is already a zero for [itex]p[/itex] of multiplicity at least [itex]2[/itex]). Anyways, what if the point [itex]a[/itex] where [itex]p'(a)=p''(a)[/itex] and [itex]p(a)\neq 0[/itex] is not in the interval between two roots (is smaller than the smallest root or larger than the largest root)?
Where are you using the assumption [itex]p(a)\neq 0[/itex]? The problem statement is false if you don't assume this.
I agree my argument was badly expressed and hard to follow, I did not like it.
Now I think we could just say:
Between any two real roots of p there must be a turning point. (1)
Every turning point is a root of p' - but not every root of p'is a turning point, in particular double root of p' is not a turning point.
p has a total of n roots. To show it has two nonreal roots it suffices to show that with the given conditions it cannot have n real roots.
So since p' has a maximum of (n - 1) real roots but two of them are not turning points, it must have maximum of (n - 3) turning points.
Therefore by (1) p has a maximum of (n -2) real roots.
Therefore p has at least two nonreal roots.
In the case that p as well as its first and second derivative are zero at x = a, we have of course a triple real root and can have a total of n real roots and no nonreal ones. We can say this possibility is allowed for in the argument because in this case there is no 'between' as required by (1).
Still a bit of a mouthful. However more obvious and memorable so the way I would want to think I would say than time traveller123's.