Can I use Newton's method to obtain smallest positive real solution of a quartic?

Question

I am familiar with Newton's method, but I'm not sure what convergence guarantees there are for this situation:

I have a quartic in real coefficients $Ax^4 + Bx^3 + Cx^2 + Dx + E = 0$, and I need to obtain the smallest positive real-valued solution.

Is Newton's method guaranteed to find the smallest positive real solution, if it's set up correctly? Is a starting guess of zero sufficient? Should I be using another method?

Answer 1

If you look for the smallest positive zero of equation $$f(x)=Ax^4 + Bx^3 + Cx^2 + Dx + E $$ $$f'(x)=4Ax^3+3B x^2+2Cx+D$$ $$f''(x)=12A x^2+6Bx+2C$$ by Darboux-Fourier theorem, you will not have any overshoot of the solution if and only if $$f(x_0)\times f''(x_0) >0$$ For your case, using $x_0=0$, this would imply $CE >0$.