Maximum and Minimum of a function
Bridge Course, August 2012

     Intuition: For all the points near , if , then  is said to have a local maximum (relative maximum) at . Similarly, For all the points near , if , then  is said to have a local minimum (relative minimum) at . Both maximum and minimum are called the extrema or turning points.

               (a) f with a maximum at x = 0                      (b) f with a minimum at x = 0 

         (c) f with a maximumat x = 1

Figure 1: Examples

     Note that: From the above, we can see that if the maximum or minimum occurs, the derivative at a point, say c, is zero or does not exist. That is, if the local extrema of f(x) occur, f′(c) = 0 or f′(c) does not exist. If there is a graph, we can easily know if the point is maximum or minimum. However, if there is no graph, what should we do?

     Definition: If f is a function with continuous first and second derivative at x = c and f′(c) = 0, in order to check if the point, c, is maximum or minimum, we have to check the second derivative of f(x) at c. If , c is a maximum point. Otherwise, if , c is a minimum point. Sometimes we may not be able to use the second derivative to check the extrema but we need to construct a sign table of f′(x) to find them.

Examples

1.  Find the maximum and minimum points of the function .

Since

 and 

 and 

 and 

Hence, f attains local minima at x=2 and -1 and a local maximum at x=1. Also see figure below.

                                                    Figure 2:

2.  If , find the local maximum and minimum points of f, if any.

Since  and

Hence, f does not have any local maximum and minimum points.

                                                                         Figure 3:

3.  If , find the local maximum and minimum points of f, if any.

     Since

For all real values x,  and f'(x) is not defined at x=2. Hence the local extrema may occur at x=2. Now, we need to construct a table to check the sign change of f'(x) for the point near x=2.

Hence,the maximum point is x = 2 and the value is f(2)=1 and no minimum points.

                                                                    Figure 4:

Exercises
Determine whether the following functions have turning point(s), and find it/them if yes.

Exercises added @ 27th Aug, 2012.

1. Find the global maximum and minimum for the function in the specified interval
1. 
2. 
2. When we combine two resistors with resistance  and  in parallel, the resulting resistance is .
   Assume that  is constant, and  can be varied in the interval . Show that  is a monotonic increasing function as a function of . Find the maximum and minimum value of . 
3. In the graph of function , 
   find the coordinates of 
1. of the point where the function is undefined,
2. of the inflexion point shown in red, 
3. of the global maximum.
4. Determine where the function  is increasing most rapidly and least rapidly.
5. Find all local maxima and local minima of . Justify your answer by either the first-derivative test or the second-derivative test.
6. If c is a nonzero constant, find all critical points of the function  defined for all nonzero x.
   Use the second-derivative test to show that if c is positive, then the graph has a local minimum, and if c is negative then the graph has a global maximum.