67 total
By the end of these notes, you should be able to:
Note: Points of inflexion are not part of this syllabus. You only need to deal with maxima and minima.
When you draw the graph of a curve, the curve goes up and down. At certain special points, the curve is neither going up nor going down — it is perfectly flat for just a moment. These are called stationary points (also called turning points).
At a stationary point, the gradient of the curve is zero.
Gradient means the steepness of the curve at a particular point. It is measured by the first derivative, dxdy.
The rule to find a stationary point:
dxdy=0
So to find stationary points, you differentiate the function, set the result equal to zero, and solve for x.
There are two types of stationary points you need to know:
A maximum point is the "peak" of a curve — like the top of a hill. The y-value at this point is higher than all the nearby points around it.
What happens to the gradient near a maximum point:
| Position | Gradient (dxdy) | Direction of curve |
|---|---|---|
| Just left of the maximum | Positive (+) | Going up |
| At the maximum | Zero (0) | Flat |
| Just right of the maximum | Negative (−) | Going down |
So the gradient goes: positive → zero → negative
Think of it like climbing a hill: you go up, reach the top (flat), then come down.
A minimum point is the "valley" of a curve — like the bottom of a dip. The y-value at this point is lower than all the nearby points around it.
What happens to the gradient near a minimum point:
| Position | Gradient (dxdy) | Direction of curve |
|---|---|---|
| Just left of the minimum | Negative (−) | Going down |
| At the minimum | Zero (0) | Flat |
| Just right of the minimum | Positive (+) | Going up |
So the gradient goes: negative → zero → positive
Think of it like going into a valley: you go down, reach the bottom (flat), then come up.
Sign in to view full notes