How to find and classify stationary points (maximum point, minimum point or turning points) of curve.

To find the stationary points of a function we must first differentiate the function. The derivative tells us what the gradient of the function is at a given point along the curve. Therefore, should we find a point along the curve where the derivative (and therefore the gradient) is 0, we have found a "stationary point".

This, however, does not give us much information about the nature of the stationary point. We can begin to classify it by taking the second derivative and substituting in the coordinates of our stationary point. Should the value of this come out to be positive then we know our stationary point is a minimum point, if the value comes out to be negative then we have a maximum point and if it is 0 we have to inspect further by taking values either side of the stationary point to see what's going on!