# Thread: find how many solutions this ln polynomial equation has..

1. ## find how many solutions this ln polynomial equation has..

$
2x^3lnx+x^2-4x+a=0
$

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

??

2. Let $f(x)=2x^3\ln x+x^2-4x+a, \ f0,\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.

$f'(x)=6x^2\ln x+2x^2+2x-4$

$f'(x)=0\Rightarrow x=1$ and f' is increasing so it has an unique solution.

$\lim_{x\searrow 0}f(x)=a, \ f(1)=a-3, \ \lim_{x\to\infty}f(x)=\infty$

If $a\in(-\infty,0]$ then the Rolle's sequence is $- \ - \ +$ and f has a solution $x_1\in(1,\infty)$

If $a\in(0,3)$ then the Rolle's sequence is $+ \ - \ +$ and f has two solutions $x_1\in(0,1), \ x_2\in(1,\infty)$

If $a=3$ then the Rolle's sequence is $+ \ 0 \ +$ and f has the solution $x=1$

If $a>3$ then the Rolle's sequence is $+ \ + \ +$ and f has no real solution.