What is the value of k if equation $x^3-3x^2+2=k$ has three real roots and if one real root? I'd like help understanding this question graphically actually. My book finds the derivative of f(x)=$x^3-3x^2+2$ and gets 3x(x-2), thus finding maxima and minima of f(x) as 2 and -2. Then it just says that for three real roots if k belongs to [-2,2] and one real root if k belongs to $(-\infty,-2)$U$(2,\infty)$.Would someone please help me understand this in a graphical,intuitive manner without using discriminant?

The idea here is that after you calculate your minimum and maximum values, which are $2$ and $0$, respectively, for $x$ and $-2 - K$ and $2 - K$ for $f(x)$, you should try to draw the thing.

Now, for you to have 3 solutions, you need $2 - K \ge 0$ and $-2 - K \leq 0$
Solving this gives you the answer
