Number of real solutions of a cubic equation without using derivatives The problem is to find the number of real solutions of a cubic equation. This exercise is in a book, in the chapter about functions, limits and continuity. This chapter is before the chapter about derivatives therefore I assume that they cannot be used to solve the exercise.
$$f(x) = x^3-9x^2+24x-17 = 0$$
Factoring it as $x(x^2-9x+24)-17$ and since $x^2-9x+24$ is always positive we can conclude that there are no solutions in $[-\infty,0]$. Since $x^2-9x+24$ is monotonically increasing in $[4.5,+\infty]$ then $f(x)$ also is. Given that $f(4.5) < 0$ we can conclude that there is an unique solution in $[4.5,+\infty]$. How to study the function in $[0,4.5]$?

Easiest, I think, to just look for sign changes.  We note that $$f(1)=-1\;\;f(2)=3\;\;f(3)=1\;\;f(4)=-1\;\;f(5)=3$$
It follows that there are real roots in each of the intervals $[1,2],\;[3,4],\;[4,5]$.  As a cubic can't have more than three roots, the answer is $\fbox 3$.
