Page 108 - Calculus Workbook For Dummies

92 Part III: Differentiation

Q. Use the first derivative test to determine 4. Test a number from each of the four
the location of the local extrema of regions, noting whether the results are
5
3
g x = 15 x - x . See the following figure. positive or negative.
^ h
Note that if you use round numbers like
y
200 0, –10, or 10, you can often do the arith-
metic in your head.
2 4
5 -
g - 10 = 45 - 10 - ^ 10 = - 45 ,500
l ^
^
h
h
h
2 4
^
1 =
l ^
g - h 45 - h 5 - 1h = 40
1 - ^
x
2 4
^ h
l ^ h
-5 5 g 1 = 45 1 - ^ = 40
5 1h
2 4
g 10 = 45 10 - ^ = - 45 ,500
l ^
^
h
5 10h
h
5. Draw a “sign graph.” Take your number
line and label each region — based on
200
your results from Step 4 — positive
(increasing) or negative (decreasing).
A. The local min is at (–3, –162), and the See the following figure.
local max is at (3, 162).
decreasing increasing increasing decreasing
1. Find the first derivative of g using the – + + –
power rule.
3
g x = 15 x - x 5 -3 0 3
^ h
2
g x = 45 x - 5 x 4 This sign graph tells you where the func-
l ^ h
tion is rising or increasing and where it is
2. Set the derivative equal to zero and
solve for x to get the critical numbers falling or decreasing.
of g. 6. Use the sign graph to determine
whether there’s a local minimum, local
4
2
45 x - 5 x = 0
maximum, or neither at each critical
2
2
x 9 -
5 _ x i = 0 number.
2
x =
x 3 - h
^
5 ^ x 3 + h 0
Because g goes down on its way to x = –3
and up after x = –3, it must bottom out
2
5 x = 0 or 3 - x = 0 or 3 + x = 0 at x = –3, so there’s a local min there.
Conversely, g peaks at x = 3 because it
x = 0 , , 3 or - 3
rises until x = 3, then falls. There is thus a
If the first derivative were undefined for local max at x = 3. And because g climbs
some x-values in the domain of g, there on its way to x = 0 and then climbs fur-
could be more critical numbers, but ther, there is neither a min nor a max at
2
4
because g x = 45 x - 5 x is defined for x = 0.
l ^ h
all real numbers, 0, 3, –3 is the complete 7. Determine the y-values of the local
list of critical numbers of g.
extrema by plugging the x-values into
Remember: If f is defined at a number c the original function.
and the derivative at x = c is either zero 3 5
3 - -
3 =
^
^
or undefined, then c is a critical number g - h 15 - h ^ 3h
of f. = - 162
3. Plot the three critical numbers on a g 3 = 15 3 - ^h 3 3h 5
^ h
^
number line, noting that they create
four regions (see the figure in Step 5). = 162
So the local min is at (–3, –162), and the
local max is at (3, 162).