Graphs of Functions and Derivatives 117
Example 2
Using a calculator, find the values ofxat which the graph ofy=x^2 exchanges concavity.
Entery 1 =x∧ 2 ∗e∧xandy 2 =d(y1(x),x, 2). The graph ofy2, the second derivative
ofy, is shown in Figure 7.2-21. Using the [Zero] function, you obtainx=− 3 .41421 and
x=− 0 .585786. (See Figures 7.2-21 and 7.2-22.)
[–4,1] by [–2,5]
Figure 7.2-21
++–
–3.41421 –0.585786
x
f
f ′′
Concave
upward
Concave
downward
Concave
upward
Change of
concavity
Change of
concavity
Figure 7.2-22
Thus,fchanges concavity atx=− 3 .41421 andx=− 0 .585786.
Example 3
Find the points of inflection off(x)=x^3 − 6 x^2 + 12 x−8 and determine the intervals where
the functionfis concave upward and where it is concave downward.
Step 1: Find f′(x) andf′′(x).
f′(x)= 3 x^2 − 12 x+ 12
f′′(x)= 6 x− 12
Step 2: Setf′′(x)=0.
6 x− 12 = 0
x= 2
Note that f′′(x) is defined for all real numbers.
Step 3: Determine intervals.
The intervals are (−∞, 2) and (2,∞).