Finding number of roots using Rolle's Theorem, and depending on parameter 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?

You can use the discriminant of the cubic function to find the nature of the roots. The general cubic equation is
$$ax^3+bx^2+cx+d=0$$
so in your case
$$a=1,\ b=3,\ c=-m,\ d=5$$
In general, the discriminant is
$$\Delta=18abcd-4b^3d+b^2c^2-4ac^3-27a^2d^2$$
You can substitute in your values and get a cubic expression in $m$. Then your equation has


*

*three distinct real roots, if $\Delta>0$.

*one or two distinct real roots, if $\Delta=0$.

*one real roots (and two complex roots), if $\Delta<0$.


To really finish your answer, you would need to distinguish one or two real roots in the second case. You may also want so solve for $m$ in each (in)equality, to give simpler conditions on $m$. Since you asked for "suggestions" and "where to begin," I'll leave it at that.
Note that this did not need Rolle's theorem. Do you really need to use it?
