### Concavity criteria. How to find a point of inflexion.

Function is concave up where the second derivative is positive or the first derivative is increasing.

Function is concave down where the second derivative is negative or the first derivative is decreasing.

How to find a point of inflexion. By definition it’s a point where a curve (function) changes concavity.

Find the second derivative and equate it to zero. Solve obtained equation with respect to x. Hence, you have found a point or points where the second derivative is zero. These point or points are candidates to be a point of inflexion. It’s necessary condition but not enough. You should check whether the second derivative changes sigh passing through your point (candidate). You should take some point to the left and calculate the value of the second derivative at that point. And the value of the second derivative at some point to the right. If you get results of different sigh, it means that the second derivative changes sigh, hence it’s a point of inflexion.

Or to prove that a given point (where the second derivative is equal to zero, our candidate) is a point of inflexion you can find the third derivative and calculate the value of the third derivative at that point (candidate). If the result is not zero, hence it’s a point of inflexion. The second method is not given in IB Math books but may be simpler and is acceptable on IB Math exam because I met it in a HL markscheme of IB Maths past papers.

Or you can show that the first derivative is increasing to the left and decreasing to the right or vice versa.

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