d²y/dx² = 0 does not necessarily mean it's a point of inflection. You should always use the first derivative method to check if you get d²y/dx² = 0

For example,

When y = x⁴, dy/dx = 4x³ and d²y/dx² = 12x.
When x = 0, dy/dx = 0 and d²y/dx² = 0.
One might think that the point where x = 0 is an inflection point. However, it is actually a minimum point when you plot the graph. Therefore, the previous conclusion would be erroneous.


However if it is first known or given that a point on the curve is an inflection point, then d²y/dx² will have to be 0

Interesting, I did not think much about this one.

It is only an inflection point when the concavity changes at that point.


Eg In her previous question where there was a graph in the form y = x³ + d, the curve changed from concave down (d²y/dx² is negative) to inflection at x = 0 and then to concave up (d²y/dx² is positive) . Therefore it was an inflection point.

Unfortunately in O levels the differences between inflection point and stationary point are not clearly stated by many schools and teachers. Many come to the wrong conclusion that all inflection points are stationary points and all inflection points have dy/dx = 0

Thank you for your help, I sometimes forget that d²y/dx² = 0 isn't always a inflection point.

You're welcome