Hello, ignis!
How i can prove that third-degree polynomial has a point of inflection?
I assume you mean a third-degree polynomial function.
The general cubic function is: .
The first derivative is: .
The second derivative is: .
A point of inflection occurs where
. .
Therefore, there is an inflection point at: .
It should be noted that the condition f''(x) = 0 is a necessary but not sufficient condition. f(x) will only have a point of inflection if f''(x) = 0 AND f'(x) has a turning point at the value of x given by the solution to f''(x) = 0.
Consideration of the graph of f(x) = x^4 illustrates why ....