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) is continue for every x and for high values of x, positives or negatives, as ax^4 weighs relatively more, F(x) will assume very high values. This also means that the general form of F(x) is like a U, with the other terms of the polinomial expression being more influencial in the low range of x.
a) To have a line tangent to two different points the graph of F(x) should pass from concave to convex and from convex to concave, which means that it should have two inflection points.
This will occur when F''(x)=0, that is practically when
9b2>24acb) The above inequality is the condition for F(x) to have two inflections points. So:
When does F(x) have a line which is tangent to it at two different points? When it has two distinct inflection points.
When does F(x) have two distinct inflection points?When it has a line which is tangent to it at two different points.

Posted by armando
on 20160416 17:40:05 