When finding the turning points of a curve, how can I tell if it is a maximum, minimum or a point of inflection?
To find the turning points of a curve one must find the values of x which satisfy dy/dx = 0. To further determine what type of turning point this is you need to compute the second derivative with respect to x, d2y/dx2. A maximum corresponds to a negative value of d2y/dx2, a minimum corresponds to a positive value of d2y/dx2 and a point of inflection corresponds to d2y/dx2 = 0. This becomes more intuitive when shown graphically, d2y/dx2 can be considered as the rate of change of the gradient of the tangent to the curve, so a maximum point will have a positive gradient go to a negative gradient, i.e. a negative rate of change of the gradient with respect to x. Similarly a minimum has a negative gradient go to a positive gradient, which is a positive rate of change. Finally a point of inflection is where the curve becomes flat, so the rate of change of the gradient is 0 as the gradient is at this point is 0.
