1. Stationary Point Notes

2. The stationary points of a curve are the points where the gradient is zero
A local maximum
A local minimum x
The word local is usually omitted and the points called maximum and minimum points.
e.g.

3. e.g.1 Find the coordinates of the stationary points on the curve
Solution: or
The stationary points are (3, -27) and ( -1, 5)

4. For a max we have
The opposite is true for a minimum
At the max
On the right of the max
On the left of the max
Calculating the gradients on the left and right of a stationary point tells us whether the point is a max or a min.
Determining the nature of a Stationary Point

5. At the max of the gradient is 0, but the gradient of the gradient is negative.
The gradient function is given by
e.g. Consider
Another method for determining the nature of a stationary point.

6. The notation for the gradient of the gradient is
“d 2 y by d x squared”
At the min of
The gradient function is given by
the gradient of the gradient is positive.

7. The gradient of the gradient is called the 2nd derivative and is written as

8. e.g. Find the stationary points on the curve and distinguish between the max and the min.
Solution:
Stationary points: or
We now need to find the y-coordinates of the st. pts.

9. max at
min at
At ,
To distinguish between max and min we use the 2nd derivative,

10. SUMMARY
To find stationary points, solve the equation
maximum
minimum
Determine the nature of the stationary points
either by finding the gradients on the left and right of the stationary points
or by finding the value of the 2nd derivative at the stationary points