122 STEP 4. Review the Knowledge You Need to Score High
- Domain: all real numbers; Range: all real numbers
- No symmetry
- Relative maximum: (1.2, 2.03)
Relative minimum: (0, 0)
Points of inflection: (−0.6, 2.56) - No asymptote
- f(x) is decreasing on (−∞, 0], [1.2,∞) and increasing on (0, 1.2).
- Evaluating f′′(x) on either side of the point of inflection (−0.6, 2.56)
d
(
−x∧
(
5
3
)
+ 3 ∗x∧
(
2
3
)
,x,2
)
x=− 2 → 0. 19
d
(
−x∧
(
5
3
)
+ 3 ∗x∧
(
2
3
)
,x,2
)
x=− 1 →− 4. 66
⇒ f(x) is concave upward on (−∞,− 0 .6) and concave downward on (− 0 .6,∞). (See
Figure 7.3-2.)
[–2,4] by [–4,4]
Figure 7.3-2
Example 2
Using a calculator, sketch the graph of f(x)=e−x^2 /^2 , indicating all relative minimum and
maximum points, points of inflection, vertical and horizontal asymptotes, intervals on which
f(x) is increasing, decreasing, concave upward, or concave downward.
- Domain: all real numbers; Range (0, 1]
- Symmetry: f(x) is an even function, and thus is symmetrical with respect to they-axis.
- Relative maximum: (0, 1)
No relative minimum
Points of inflection: (−1, 0.6) and (1, 0.6) - y=0 is a horizontal asymptote; no vertical asymptote.
- f(x) is increasing on (−∞, 0] and decreasing on [0,∞).
- f(x) is concave upward on (−∞,−1) and (1, ∞); and concave downward on
(−1, 1).
(See Figure 7.3-3.)