Page 128 - Calculus Workbook For Dummies

112 Part III: Differentiation

e Use the second derivative test to analyze the critical numbers of the function from problem 2,
x 2 2 2
h x = + cosx - . Local max at x = - ; local min at x = .
^ h
2 2 2 2
1. Find the second derivative.
x 2
h x = + cosx -
^ h
2 2
1
h x = - sinx
l ^ h
2
h x = - cosx
m ^ h
2. Plug in the critical numbers (from problem 2).
π π 3 π 3 π
hm c m = - cos hm c m = - cos
4 4 4 4
2 2
= - =
2 2
2
You’re done. h is concave down at x = - 2 , so there’s a local max there, and h is concave up
2
at x = , so there’s a local min at that x-value.
2
(In problem 2, you already determined the y-values for these extrema.) h is an example of a
function where the second derivative test is quick and easy.
2
3
1
f Find the local extrema of f x = - 2 x + 6 x + with the second derivative test. Local min at
^ h
(0, 1); local max at (2, 9).
1. Find the critical numbers.
2
3
f x = - 2 x + 6 x + 1
^ h
2
f l ^ h 6 x + 12 x
x = -
2
0 = - 6 x + 12 x
x x -
0 = - 6 ^ 2h
x = , 0 2
2. Find the second derivative.
2
x = -
f l ^ h 6 x + 12 x
x = -
f m ^ h 12 x + 12
3. Plug in the critical numbers.
0 = -
f m ^ h 12 0 + 12 f m ^ h 12 2 + 12
2 = -
^ h
^ h
= 12 _ concave up : mini = - 12 ^ concave down: maxh
4. Determine the y-coordinates for the extrema.
3 2 3 2
^
^
f 0 = - ^h 2 0 + ^h 6 0 + 1 f 2 = - ^h 2 2 + ^h 6 2 + 1
h
h
= 1 = 9
So there’s a min at (0, 1) and a max at (2, 9). f is another function where the second derivative
test works like a charm.