Let F(x) be the fourth degree polynomial ax^4 + bx^3 + cx^2 + dx + e, with a>0.
When does F(x) have a line which is tangent to it at two different points?
When does F(x) have two distinct inflection points?
F(x) should satisfy two conditions (to have a line tangent common at two different points x1 and x2):
The first condition implies F'(x1)-F'(x2)=0 which result in:
To work out the second condition is useful:
- x1^3-x2^3=(x1-x2)*(x1^2+x1*x2+x2^2). The bold expression can be worked from 
Anyway there is some calcutation involved, it's easy to get lost with potences and signs... At the end I obtain:
and using 
From  and  it's possibile to get x1 and x2 as an (a,b,c, dependent expression)
Apologize. Expresions  and  wrong...
Anyway: between x1 and x2 it would be an inflection point x3. For that point
F''(x3)=0 --> 12ax3^2+6bx3+2c=0 -->
x3=(-3b +/- sq (9b^2-24ac)/12a which is real for 9b^2>24ac
With that condition we should have two inflection points (but only one of them betwen x1and x2) and there is a line tangent to F(x) in two different points.
Edited on April 15, 2016, 2:25 pm
Posted by armando
on 2016-04-15 06:38:13