1. Let a function have the derivative on a contiguous domain D(f '). Then is a function on D(f ') and, as such, may again have the derivative. If so, it is denoted and called the second derivative of Sometimes we also use the denotation Examples

2. In a similar way, we can define the third, fourth, etc. derivative of a function if such derivatives exist. For a function we use the following notation the third derivative of f the fourth derivative of f the n-th derivative of f

3. If is defined on an interval and then is increasing on If then is decreasing on

4. Bending-up and bending-down curves Let be a continuous function defined on an interval Assume that exist for. We view the second derivative as the rate of change of the slope of the curve over the interval. If the second derivative is positive in the interval, then the slope of the curve is increasing and we interpret this as meanig that the curve is bending up. If the second derivative is negative, we interpret this as meaning that the curve is bending down. a bb a bending upbending down

5. Inflexion points A point where a curve changes its behaviour from bending up to down (or vice versa) is called an inflexion point. If the curve is the graph of a function whose second derivative exists and is continuous, then we must have at that point.

6. bending up bending down inflexion point

7. Maximum and local maximum of a function Let a function be given with a domain. For a we say that c is the point of a maximum of the function if If a neigbourhood exists such that we say that has a local maximum at c.

8. Minimum and local minimum of a function Let a function be given with a domain. For a we say that c is the point of a minimum of the function if If a neigbourhood exists such that we say that has a local minimum at c.

9. minimum maximum points of local maximum points of local minimum

10. Let a continuous function be defined on an interval. Let c be a point of maximum (minimum) of on. Then has a local maximum (minimum) at c or c is one of the extreme points a and b a b a b

11. If a function is to have a local maximum at a point c, then there must be a neighbouhood of c such that  is increasing in and decreasing in Thus we have This is only possible if either of the derivative of at c does not exist. For a local minimum, we may use a similar reasoning

12. A sufficient condition for a local minimum/maximum If, for a function, we have then has a local maximum at c. If then has a local minimum at c.

13. With k = 1 we have a particular case of this rule : and

14. Example Find minima and maxima of the function Maximum at -1.

15. Example Verify if has a local maximum or minimum We have and so We conclude that f (x) has a local minimum at 0.

16. Example Find the local minima and maxima of the function x y

17. We have Thus at 0 does not exist which means that has no derivative at 0. Since it is increasing on the left of 0 and decreasing on the right, we conclude that it has a local maximum at 0.

18. Find the maxima and minima of the function on the interval [  2, 1.5] Since local minima or maxima can only be reached at points where the first derivative is zero and minima or maxima can also be reached at the extreme points, the only canidates for maxima and minima are the points  2,  1,0,1,1.5. Calculating the values at these points we can verify that minima are reached at the points  1 and 1 while the maximum is reached at the point  2