Classifying stationary points

To find the stationary points of y=f(x), first differentiate with respect to x and then set equal to 0. This gives the points where the tangent to the curve is horizontal.

However there are three possible shapes, as shown in the table.

To find which is which, look at the sign of the derivative on each side of the stationary point. Alternatively, you can differentiate again, to get the second derivative, written d²y/dx².

This tells you how rapidly the slope of the tangent is changing at a point.

	Maximum	Minimum	Inflection
sign of dy/dx	changes from + to -	Changes from - to +	same sign, + + or - -
sign of d²y/dx²	negative	Positive	0

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