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5 Maximum and Minimum of a function

Page history last edited by Kenneth Shum 11 years, 10 months ago

Maximum and Minimum of a function
Bridge Course, August 2012

 

     Intuition: For all the points near Formula, if Formula, then Formula is said to have a local maximum (relative maximum) at Formula. Similarly, For all the points near Formula, if Formula, then Formula is said to have a local minimum (relative minimum) at Formula. 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 Formula, c is a maximum point. Otherwise, if Formula, 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 Formula.

                    Formula

 

Since

Formula and Formula

Formula and Formula

Formula and Formula

 

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

                                                    Figure 2: Formula

 

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

 

          Formula

          Formula

 

Since Formula and Formula

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

                                                                         Figure 3:Formula

 

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

     SinceFormula

 

For all real values xFormula 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.

 

x
x < 2
x = 2
x > 2
f′(x) +
undefined

f(x)

max.pt

 

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

                                                                    Figure 4: Formula

 

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

  1.  Formula
  2.  Formula
  3.  Formula
  4.  Formula
  5.  Formula
  6.  Formula

 

Exercises added @ 27th Aug, 2012.

 

  1. Find the global maximum and minimum for the function in the specified interval
    1. Formula
    2. Formula
  2. When we combine two resistors with resistance Formula and Formula in parallel, the resulting resistance is Formula.
    Assume that Formula is constant, and Formula can be varied in the interval Formula. Show that Formula is a monotonic increasing function as a function of Formula. Find the maximum and minimum value of Formula
  3. In the graph of function Formula
    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 Formula is increasing most rapidly and least rapidly.
  5. Find all local maxima and local minima of Formula. 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 Formula 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.

 

 

 

 

 

 

 

 

 

 

Comments (1)

Kenneth Shum said

at 5:30 pm on Aug 27, 2012

At the end of the class of 26th Aug, I gave a proof of the question asked by Prof. Chandra Nair in the previous class, that the infinite sum of reciprocal of the primes diverges. The proof I gave relies on the inequality that
-log(1-x) is less than or equal to x+2x^2 for x between -0.5 and 0.5.

Since the divergence of this series is a famous mathematical result, there are many proofs. Some of the other proofs can be found in wikipedia:
http://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes

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