A-LEVEL MATHEMATICS Topic : Maxima and Minima – The Second Differential Criterion

If a curve is sloping upis positive. and if a curve is sloping down thenis negative

The graph on the left hasincreasing – it goes from negative to zero to positive. This means that the gradient ofis positive.
The graph on the right hasdecreasing – it goes from positive to zero to negative. This means that the gradient ofis negative.
In both cases at the actual turning point (maximum or minimum) the gradientis zero. To find and classify the turning points we first differentiate and setequal to zero. We solve this equation to find the x values of the turning points, then differentiateto findand put the values we have found into this expression. If the value we obtain here is positive then we have found a minimum forIf the value we obtain is negative then we have found a maximum forIf we need to find the– coordinate too we can substitute the– values of the minimum into the original expression for
To summarise:
To find a turning point solve for
To classify a turning point, put thevalues of the turning point into the expression for
If this value is positive, we have a minimum, and if it is negative we have a maximum. To find the – value of the turning point, substitute the– values of the turning point into the expression for
Example. Find and classify the turning points of
Solve
so the coordinates of the turning point are
Therefore this is a minimum.



Example. Find and classify the turning points of
Solve

When
When
Attherefore this is a minimum.
At therefore this is a maximum.