1.(10 points) a)  State either the Extreme Value Theorem or the Mean Value Theorem.
b)  Tell why the theorem you stated in (a) is considered to be a useful theorem.

2.(10)  A landscape architect plans to enclose a 3000 square-foot rectangular region in a garden.  The plan calls for planting shrubs along three sides; they cost $25/foot.  There will be a fence on the remaining side costing $20/foot.   Find the dimensions that minimize the total cost.  Be sure to organize your work so that it can be evaluated.

3.(15)  Decide if the following statements are true or false.  Provide a reason for your answer.
_____ a.  If f ¢¢(x₀) = 0 then f has an inflection point at x = x₀.
_____ b.  If f ¢ is increasing on an interval, then f is concave up there.
_____ c.  If f is decreasing and concave down for -¥ < x < ¥, then f must have at least one root.

4.(25)  The graph show the derivative, f ¢, of a mystery function f.  Answer the following questions and provide a reason for each answer (a)-(d). graph

a)  What are the critical points of f?
 ANS:  
 REASON:  
b)  What are the inflection points of f?
 ANS:  
 REASON:  
c)  On what intervals is f decreasing?
 ANS:  
 REASON:  
d)  For what x in the range 0 £ x £ 100 does f(x) have a maximum value?
 ANS:  x =
 REASON:  
e)  Assume that f(25) = 50.  Evaluate f(50).  Show your work.

5.(5)  Suppose that at a production level of 2000 for a given product, marginal revenue is $4 per unit and marginal cost is $3.25 per unit.  Do you expect maximum profit to occur at a production level above or below 2000?  Explain your answer in a few sentences.

6.(35)  Consider the function f(x) = x² e^-2x defined on the whole line -¥ < x < ¥

Find and simplify the derivatives;  show your work; check your answers.
a)  f ¢(x) =
b)  f ¢¢(x) =

USE CALCULUS to answer the following questions.  No credit without showing your work.  Some words of explanation of your work is necessary.
c)  Find the interval(s) on which f is increasing.
d)  Find the interval(s) on which f is concave up.
e)  Find the local extrema (if any) for f.  Be sure to show your calculus work.
Answer:  Local max at x =  _____; value of local max _________
 Local min at x =  _____; value of local min _________
f)    Must f have a global maximum?  Why?
Does f have a global maximum?
g)  Suppose we consider f only on the interval [-0.5,3].  Why must f have a global maximum on [-0.5,3]?    Where does the global max occur?  What is the global max there?