[Math] Finding number of roots using Rolle’s Theorem, and depending on parameter

Question

I need to count the number of real solutions for $ f(x) = 0 $ but I have an $m$ in there.
$$ f(x) = x^3+3x^2-mx+5 $$
I know I need to study $m$ to get the number of roots, but I don't know where to begin. Any suggestions?

Answer 1

Taking the derivative of the function: $$f'(x)=3x^2+6x-m$$ If the function $f$ has two roots with $x=a$ and $x=b$, then $$f(a)=0=f(b)$$ Since $f(x)$ is continuous and differentiable, by Rolle's theorem, we have that there exists at least one $c \in (a,b)$ such that $$f'(c)=0$$ Namely, $$f'(c)=3c^2+6c-m=0$$ And here is where you begin to consider how the roots exist depend on the parameter.