how to find how many times a line crosses a curve?

Question

Given $f(x)=\frac{x^3}{3}-4x+1$, for what $m$ values the equaition $f(x)=m$ has:

(1): 1 solution

(2): 2 solutions

(3): 3 solutions

Can this question be solved without sketching a graph? If so, how? Thanks!

Answer 1

$$f(x)=x^3-12x+3-3m\Rightarrow f'(x)=3x^2-12=0 \rightarrow x=\pm 2$$ $$f''(2)=12, f''(-2)=-12.$$ Then $$f_{max}=f(-2)=19-3m, f_{min}=-13-3m$$ (1) Three distinct root if: $f_{max}>0$ and $f_{min} <0 \Rightarrow -13/3 <m <19/3.$

(2) Two distinct roots if $m= 19/3, -13/3$

(3) Other wise only one real root.