## Maxima and Minima - The Second Derivative Criterion

If a curve is sloping up is positive. and if a curve is sloping down then is negative The graph on the left has increasing – it goes from negative to zero to positive. This means that the gradient of is positive.

The graph on the right has decreasing – it goes from positive to zero to negative. This means that the gradient of is negative.

In both cases at the actual turning point (maximum or minimum) the gradient is zero. To find and classify the turning points we first differentiate and set equal to zero. We solve this equation to find the x values of the turning points, then differentiate to find and put the values we have found into this expression. If the value we obtain here is positive then we have found a minimum for If the value we obtain is negative then we have found a maximum for If 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 the values 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 At therefore this is a minimum.

At therefore this is a maximum. 