©Zarestky
Math 142 Quiz 6
3/24/2011
NAME (print): ____KEY______________
Section (circle one):
1.
508
509
(1 pts) The function f (x) is continuous for all real numbers and f !(3) = 0 and f !("1) = 0 . Based on the
following table of function values, what are the absolute extreme values of f (x) on the interval [0, 4]?
x
f (x)
–1
25
0
19
3
0
4
2
x = −1 is not in the interval. Correct responses correspond to f (x) values.
Absolute max: __19___
Absolute min: ___0____
2.
(1 pt) The function f (x) is continuous for all real numbers. If f !(2) = 0 and f !!(2) > 0 what do you
know for certain about f (2) ?
Circle one:
Local Max
Abs Max
Local Min
Abs Min
Concave up corresponds to a minimum but you cannot tell from the second derivative alone if the
minimum is an absolute minimum.
©Zarestky
3.
Math 142 Quiz 6
3/24/2011
(8 pts) The owner of a retail lumber store wants to construct a fence to enclose an outdoor storage area
adjacent to the store, using all of the store as part of one side of the area. The side of the store,
represented by the dotted line, is 100 ft long. What are the dimensions that will enclose the largest
possible area if 400 ft of fencing are used? Note: the figure below is NOT drawn to scale.
100ft
A. What are you trying to optimize? Circle one: Area Perimeter Cost
y
B. Do you want the maximum or minimum? Circle one: maximum minimum
x
C. Write an equation to represent the area of the fenced in region.
A = xy
D. Write an equation in terms of x and y to represent the amount of fencing that will be used.
400 = 2y + x + (x – 100) = 2y + 2x – 100
or 500 = 2y + 2x
E. Substitute your answer to D into your answer to C. (You will need to do some algebra.) This is the
function you must optimize.
Option 1: Solve for x in the fencing equation: x =
Then A = y (250 ! y ) = 250 y ! y 2 .
Option 2: Solve for y in the fencing equation: y =
Then A = x (250 ! x ) = 250x ! x 2
500 ! 2 y
= 250 ! y
2
500 ! 2x
= 250 ! x
2
F. Find the critical values of your answer to E.
A! = 250 " 2x = 0
x = 125
or
A! = 250 " 2 y = 0
y = 125
G. Identify the absolute maximum or minimum (as appropriate) value of the function. Use a sign chart
or the second derivative test to verify your answer.
The maximum area is A = 250(125) !1252 = 15,625 ft 2
A!! = "2 < 0 so x = 125 (or y = 125) is the location of a maximum 
H. Answer the question.
The dimensions that maximize area are 125 ft by 125 ft.