Example
At x1 we have f' (x) = 0 and f''(x) = 0. Even though f''(x) = 0, this point is not a point of inflexion. The reason is that the sign of f' (x) changes from negative to positive.
At x2, we have f' (x) 0 and f''(x) = 0. But, x2 is not a stationary point, rather it is a point of inflexion. This because the concavity changes from concave downwards to concave upwards and the sign of f' (x) does not change; it stays positive.
At x3 we have f' (x) = 0 and f''(x) = 0. Here, x3 is both a stationary point and a point of inflexion. This is because the concavity changes from concave downwards to concave upwards and the sign of f' (x) does not change; it stays positive.
Assuming that f'(x) < 0, there are no distinct roots. Hence f''(x) = dy.
Read more about this topic: Stationary Point, Curve Sketching
Famous quotes containing the word example:
“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)