So I practicing questions from Differential Calculus and came across a problem which required me to find the number of real roots of a polynomial equation, say $f(x)=0$. I initially thought of applying Sturm's theorem but was in a dilemma as to how this question relates to Differential Calculus. I looked for a solution and came across a statement which stated
Let $f(x)=0$ have $n$ number of real roots. Then by Rolle's theorem, $f'(x)=0$ will have $n-1$ real roots and $f''(x)=0$ will have $n-2$ real roots.
Can I therefore conclude that if a polynomial function has $k$ number of real roots then it's $n$th differential coefficient $f^{(n)}(x)=0$ will have $k-n$ number of real roots? Also, if that's the case, why is there a theorem as lengthy as Sturm's theorem if one could simply determine number of real roots of an equation through differential calculus?
Why not using the Bolzano's Theorem if the function is continuous in all its domain? In this case you can use the derivative to find out ranges in which your function is monotonic. Then you can calculate the image of each 0 of the derivative through the f(x) function and, if the product of the images of two consecutive zeros is < 0 you have a zero in that range. In the case of the smallest 0 you'll have to compare it against the limit for x-> a from the right (being the left extreme of your domain) of f(x) and, same for the biggest 0 of f'(x), to be compared against limit for x-> b from the left of f(x).