Stationary Points

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There are three types of gradient a curve can have
In this case the curve is increasing
Alternatively the curve could be decreasing
However, the most interesting case is.

At this point the curve is stationary.

This allows you to find the maximum or minimum value of a function, since these always occur where the gradient is 0
Stationary points can come in 3 varieties
A Minimum
A Maximum
A Point of Inflection
Below are examples of finding the 3 types of stationary points.
To find the stationary points, differentiate and say that the result must equal 0.
Solve this equation, to find the value of x at which the stationary point occurs.
Put this value of x into y.
Therefore, the curve has a stationary point at
To find the nature of the stationary point, you look at the gradient on either side of the point. Gradient at x = -2
Gradient at x = 0
So the gradient is negative to the left of the point and positive to the right.

This can be represented by a sketch

The point is a minimum.

Differentiate and say that the result is 0.
Solve the equation to find x.
Put this value into y.
So the stationary point is at
To find the nature of the point, look at the gradient on either side of the point
The gradient is positive to the left and negative to the rightThe point is a maximum

Differentiate and say that the result is 0
Solve to find x and the corresponding y.
Look at the gradient on either side of the point.
The gradient is positive on both sides of the pointThis is a point of inflection.
Note that a point of inflection could be negative on both sides of the point.

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