How do I know if a curve is convex?

The second differential of the equation of the curve will be positive if it is convex. This is because, by definition, convex means that the gradient of a curve is increasing. To find the gradient of a curve we do the first differential of the curve as that shows us the rate of change of the initial curve. Therefore, to find whether or not the rate of the change of the initial curve is increasing (i.e. the curve is convex) we find the rate of change of the gradient by finding the differential of the equation of the gradient curve. We are therefore finding the differential of the differential and checking whether or not it is greater than 0. If it is then we know that the rate of change of the gradient of the initial curve is increasing and it is therefore convex.