Page 126 - Calculus Workbook For Dummies

110 Part III: Differentiation

/ 2 3
2
c Locate the local extrema of y = _ x - 8i with the first derivative test. Local mins at - 2 , 2 0j
`
and 2 2 0j; a local max at 0 4i.
,
,
_
`
Same basic steps as problems 1 and 2, but abbreviated a bit.
1. Find the derivative.
/ 2 3
2
y = _ x - 8i
2 2 - / 1 3 4 x
y = l _ x - 8i 2 ^ x =
h
3 3 3 x - 8
2
2. Find the critical numbers.
4 x
= 0
2
a. 3 3 x - 8
x = 0
b. The first derivative will be undefined when the denominator is zero, so
2
3 3 x - 8 = 0
x - 8 = 0
3 2
2
8
x - = 0
2
x = 8
x = ! 2 2
The critical numbers are 2 2, 0, and 2 2.
-
3. Test the values.
2 2 - / 1 3 2 2 - / 1 3
1 =
1 -
l ^
y - 10 = ^ a - 10 - 8k ` 2 $ ^ - 10hj y - h ^ a - h 8k ` 2 $ ^ - 1hj
h
l ^
h
3 3
2 - / 1 3 2 - / 1 3
= _ positivei $ negative = $ _ negativei $ negative
3 3
2 2
$
$
= positive negative = $ negative negative
3 3
= negative = positive
y 1 = negative and y 10 = positive _ What? I should do all your work?i
h
l ^
l ^ h
4. Make a sign graph (see the following figure).
decreasing increasing decreasing increasing
– + – +
0
5. Find the y-values.
/ 2 3
2
j
y = ` c - 2 2 - 8m = 0 There’s a local min at - 2 , 2 0j.
`
/ 2 3 / 2 3
2
^
y = _ 0 - 8i = - 8h = 4 There’s a local max at (0, 4).
/ 2 3
2
,
`
j
y = ` c 2 2 - 8m = 0 There’s another local min at 2 2 0j. Check out this interesting curve
on your graphing calculator.