Stationary/Turning Points How do we find them?

What are they?  Turning points are points where a graph is changing direction  Stationary points are turning points that have zero gradient  These occur at maximum and minimum points on a graph.

Max or Min?  Where are the stationary points on this graph?  Which ones are maxima and which are minima?

Finding Stationary Points  What is the gradient function equal to at a stationary point? To find coordinates of a stationary point:  Set the gradient function equal to 0 and solve the equation to find the x-coordinate (or coordinates if there is more than one)  Use the original equation of the graph to find the y-coordinate

Example  y = x 3 – 3x 2 - 9x + 6  What is the gradient function?  dy/dx = 3x 2 – 6x - 9  So 3x 2 – 6x – 9 = 0 x 2 – 2x – 3 = 0 x 2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 (x – 3)(x + 1) = 0 So x = 3 or x = -1 So x = 3 or x = -1

Example: y = x 3 – 3x 2 - 9x + 6  x = 3 or -1  Find the y-coordinates for each of these points

Example 2: y = x 2 – 4x + 5  Find the coordinates of the stationary point(s)  (2, 1)

Example 3: y = 2x 3 + 3x 2 – 12x + 1  Find the coordinates of the stationary point(s)  (1, -6) and (-2, 21)

How do we decide if a stationary point is a max or min without drawing the whole graph?  When any graph is heading towards a maximum point it has a positive gradient  When any graph is heading away from a maximum point it has a negative gradient  Can you make similar statements about a minimum point?

Summary positive zero negative positive negative zero Maximum Point Minimum Point

Max or min?  How could we decide if our points are max or min? (use what we have just found)  Try points just before and just after your stationary points and see what the gradient is.  Let’s try this with our first example….

Example  Stationary Point : (3, -21)  So following the pattern it is a MINUMUM point 03negative2Gradient X coordinate positive4

Example  Stationary point : (-1, 11)  So it is a MAXIMUM point. X coordinate 0positive-2Gradientnegative0

Try this one  y = x 3 – 12x + 3  Find dy/dx (the gradient function)  Find the coordinates of the stationary points (set the gradient function equal to 0)  Decide if each of them is a max or a min

Points of Inflection  Another type of stationary point  The graph “flattens” to give a point with horizontal gradient before continuing with the same type of slope as before  To see this more clearly…..

Points of Inflection  y = 2x 3 – x 4  Where is the gradient equal to zero?  There is a maximum here  There is a point of inflection here

How do you know which it is?  We find points of inflection in exactly the same way as we find the other stationary points, by setting the gradient function equal to zero  All we need to worry about is deciding if a stationary point is a point of inflection…

What are the gradient patterns? Points of inflection can have either of the following gradient patterns: Negative Zero Negative Positive Zero

Points of Inflection  So to decide if a stationary point is a point of inflection you have to check points either side of it in a table as before and see what the gradient pattern looks like.  Now let’s try one….

Example  y = 4x 3 – x 4  Find the gradient function  Set it equal to 0 and solve for x  Decide if your points are max, min or points of inflection  Sketch the curve