I have a question about derivative sketching. The second derivative test allows us to find a point of inflection if f'(x) = 0 and f''(x) = 0. However, for some functions like x^4, this is not the case. It starts concave up before the point, and then after the point it is still concave up, which does not make it a point of inflection. Can anyone explain this for me?

Question

I have a question about derivative sketching. The second derivative test allows us to find a point of inflection if f'(x) = 0 and f''(x) = 0. However, for some functions like x^4, this is not the case. It starts concave up before the point, and then after the point it is still concave up, which does not make it a point of inflection. Can anyone explain this for me?

Answer 1

Sal Kahn does a great job of explaining the concepts of concavity, points of inflection, and the concepts associated with this topic

https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-6a/v/concavity-concave-upwards-and-concave-downwards-intervals

https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-analytical-applications-new/ab-5-6a/v/inflection-points

clearly if y = x^4
y' = 4x^3
y'' = 12x^2 <---- This is concave upwards for all real values of x, thus we cannot have a
point of inflection for this function.