Find a set of values of the constant k for which a 4th degree polynomial has four real roots?

Question

I am having trouble in figuring out the best approach on how to start off this question. Hope somebody has an idea.

Answer 1

If for some $k$, the polynomial function $$ f(x)=3x^4+4x^3-12x^2+k$$ has the maximum possible number of four distinct roots, then between any two distinct roots, there is a local extremum, more precisely, first a minimum, then a maximum, then a minimum again (because $f(x)\to+\infty$ as $x\to \pm\infty$). Apparently, the values at the minima must be negative and at the maximum positive, and as long as we respect these sign rules, there will be four distinct real roots.

The extrema of $f$ are at the roots of $f'(x)=12x^3+12x^2-24x=12x(x-1)(x+2)$, i.e., at $x=0$, $x=1$, $x=-2$. We compute $f(-2)=k-32$, $f(0)=k$, $f(1)=k-5$, so from the preceding remarks arrive at the condition $$0<k<5. $$