From previous calculations, we understand that at any particular point, the gradient of a curve is positive when it is moving up at that point, the gradient is positive, and the gradient is negative when the curve is moving down. However, we call the curve stationary when a curve passes through a horizontal movement (gradient = 0). In other cases, inflexion points also occur.
When y' > 0, the curve is increasing
When y' < 0, the curve is decreasing
When y' = 0, the curve is stationary.
1. Find all x values for which the curve is increasing.
2. Find any stationary points on the curve .
Types of Stationary Points
The three types of stationary points are:
1. Local Minimum
Local minimum is the lowest value between a specified range of a function. The reason we say local is because it is possible that the largest and the smallest value of the whole function is at the endpoint of a graph. As seen from the graph above, the curve is decreasing on the left and increasing on the right of the minimum turning point. That is,
2. Local Maximum
Local maximum and minimum points are also called turning points, or maxima or minima.
To check whether a point is a maxima or minima, analyse the region to the left or right of the turning point in question, and derive a conclusion of the gradient of the curve.
3. Point of Inflexion
Another type of stationary points (gradient = 0) is a point of inflexion, where the curve stops increasing (or decreasing) at one point and then continued its increasing (or decreasing) pathway. Therefore, if the curve is increasing to the left side of a turning point (x, y), the curve will continue increasing to the right of the turning point. The same pattern appears when the curve is decreasing.
1. Find the stationary point on the curve and determine what type it is.
2. Find any stationary points on the curve and distinguish them.