i thought of building a taylor series around 0 for ln
but ln(0) is undefined
??
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.