4.5 The Second Derivative Test

The second derivative test is an alternative to doing the sign diagrams. It can be faster in some applications.

We can use the second derivative to examine what the first derivative is doing.

We already know that if the first derivative is equal to zero and goes from increasing to decreasing then the turning point is a maximum and from decreasing to increasing indicates a minimum.

The Second Derivative Test

If f'(c) = 0 and f''(c)>0 then the turning point is a minimum at c.

If f'(c) = 0 and f''(c)<0 then the turning point is a maximum at c.

The rationale is as follows: if f'(c) = 0 then there is a local extrema at c and the fact that f''(c)>0 indicates that the f'(x) is increasing. If f''(c)<0 then f'(x) is decreasing.

This method is better used when the second derivative is easy to calculate. (when you have negative exponents, you should probably avoid this method)

The second thing the second derivative can tell you is intervals of concavity. If f''(x)>0 on an interval then the slopes of the tangents are increasing on the interval. The shape of f(x) on that interval is said to be concave up on that interval, or like a bowl. If f''(x)<0 on an interval then the slopes of the tangents are decreasing on the interval. The shape of f(x) on that interval is said to be concave down on that interval, or like a hill. A point of inflection is the point where a curve's concavity changes.

Ex 1 Draw a sketch of the following curves. Include details such as intervals of increase/decreasing, intervals of concavity, stationary points, intercepts and points of inflection.

(a) (b)

p. 616#1 but sketch.