Stationary+Points

Stationary Points & Their Nature
Remember that functions are stationary at points where the gradient is equal to zero: f'(x)=0. At these points the curve is neither increasing or decreasing. There are three types of stationary points: Note:
 * =  ||= 1st derivative ||= 2nd derivative ||
 * = Minimum Turning Point ||= f'(x)=0 ||= f"(x)>0 ||
 * = Maximum Turning Point ||= f'(x)=0 ||= f"(x)<0 ||
 * = Point of Inflection ||= f'(x)=0 ||= f"(x)=0 ||
 * There is a subtle difference between stationary points & turning points: turning points only include minimum & maximum, while stationary points are all points where the gradient is zero (so including points of inflection as well).
 * Points of inflection are where the graph changes in concavity. They are sometimes but not always stationary points.

1. differentiate the function 2. set f'(x)=0 and solve for x 3. substitute into f(x) to find y coordinate > a. Sketch the graph & determine visually. (use your calculator) b. look at the gradient just before and just after this point. gradient approach c. Use the 2nd derivative test: differentiate the function again and substitute in the x value of the stationary point into f"(x). The sign of the 2nd derivative tells you about the concavityof the graph (also see delta p.106): if it is positive, the graph is concave up (so the point is a min), if it is negative, the graph is concave down (so point is a max), if zero, the concavity is changing (so point is one of inflection). Ex10.5 p. 112.
 * To find stationary points:**
 * [|Example]
 * Ex.10.1 p. 103
 * Approaches to determine the nature of a turning point:**