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Maxima and Minima - The Second Derivative 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 thevalues 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.

Attherefore this is a maximum.