Let 0,\infty)\to\mathbf{R}" alt="f(x)=2x^3\ln x+x^2-4x+a, \ f0,\infty)\to\mathbf{R}" />

We use the Rolle's sequence.

and f' is increasing so it has an unique solution.

If then the Rolle's sequence is and f has a solution

If then the Rolle's sequence is and f has two solutions

If then the Rolle's sequence is and f has the solution

If then the Rolle's sequence is and f has no real solution.