Mathematics 161                                  Laboratory 7                             October 15, 1992
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994

Extreme Values of Functions

In this lab we will investigate the relationships between the derivative of a function and the largest and smallest values a function assumes.

Part 1: A Hypothesis About Extreme Values

Consider the polynomial function p(x) defined below:

p[x_] := (3/5)x^5 - x^3 + 1

By plotting the graph of this function we can easily estimate where the function is increasing and where it is decreasing.  For instance, use the plot command below.

Plot[p[x], {x,-1.5,1.5}]

Based on this graph, on what x-intervals is the function increasing, and where is it decreasing?

The function value at a point where an interval of increase ends and an interval of decrease begins is called a local maximum (plural maxima).  Similarly, the value at a point where a function changes from decreasing to increasing is called a local minimum (plural minima).  To find the local extrema (maxima and/or minima) of p, we first estimate the inputs (x values) that end intervals of increase and decrease.  They are:

p(x) changes from increasing to decreasing at about x = ___________ (call this xmax).

p(x) changes from decreasing to increasing at about x = ___________ (call this xmin).

Recall, in lab 5 you made a hypothesis about the relationship between the sign of the derivative of a function and the function's intervals of increase and decrease (look at it now if you don't remember what it was).  Based on your hypothesis, complete the following sentences:

Changing from inputs just below xmax to inputs just above it, the sign of p' will

_______________________________________________________________________.

Changing from inputs just below xmin to inputs just above it, the sign of p' will

_______________________________________________________________________.

Create and print out a plot of the graph of the derivative of p and state (yes or no) whether the graph supports the sentences above.  To determine whether you have found the exact x values corresponding to the local extrema, compute the value of the derivative at your estimates.  Record Mathematica's output here.

p'[xmax]

p'[xmin]

(You must substitute in your estimates before these commands will work!)  Based on these derivative values, have you found the exact x values of the extrema?  If so, how can you tell, if not, how can you improve your estimate?

What are the local maximum and minimum of p?  (Look carefully at the definitions before you answer!)

Here is a statement that is "almost true:"

              If the derivative of a function f is zero at x = x0, then f(x0) is a local  extremum.

Explain why the statement is false.  You may find it helpful to refer to your graph of p'. 







Complete the sentence below to form a hypothesis that avoids the fallacy above.

Hypothesis 1:  If f(x0) is a local extremum, then

Part 2:  Investigating the Hypothesis

Consider now the function

Define the function in Mathematica, and plot its graph over the interval [–1, 3].  Identify the intervals of increase and decrease, and estimate the x values corresponding to local extrema.  Write your answers below:

Plot the graph of q' over [–1, 3], and indicate the value of the derivative at the points you estimated to be extrema above.  Write your estimated "values" below.

How do these results affect your hypothesis?


Modify your hypothesis to accomodate these results, and write it below:

Hypothesis 2:

The book defines a critical point of a function f as a point x = a where f '(x) = 0 or
f '(x) does not exist.  Based on your hypothesis, explain why such a point might justifiably be called critical.

Part 3: A Mysterious Function

Use the command below to enter the definition of a"mysterious" function, g(x).

<<i:\course\carducci\math161\mystery.m

Apply Hypothesis 2 to find all the local extrema of g(x) in the interval [–3,3] and classify each one as either a local maximum or a local minimum. Mathematica can't evaluate the derivative of this function, but, if you ask, the lab assistant or your instructor will show you a command that will provide you with the graph of g'(x).    Write a brief essay explaining how you identified and classified the extrema.