MA 3972-MA-Book April 11, 2018 17:21148 STEP 4. Review the Knowledge You Need to Score High
Step 3: Determine the intervals.- 1 01
The intervals are (−∞,−1), (−1, 0), (0, 1), and (1,∞).
Step 4: Set up a table.
INTERVALS (−∞,−1) X=−1(−1, 0) X= 0 (0, 1) X= 1 (1,∞)
Test Point − 2 − 1 / 2 1/2 2
f′(x) − undefined + 0 − undefined +
f(x) decr. rel. min. incr. rel. max. decr. rel. min. incr.Step 5: Write a conclusion.
Using the First Derivative Test, note that f(x) has a relative maximum atx= 0
and relative minimums atx=−1 andx=1.Note thatf(−1)=0,f(0)=1, andf(1)=0. Therefore, 1 is a relative maximum value and
0 is a relative minimum value. (See Figure 8.2-13.)[–3, 3] by [–2, 5]
Figure 8.2-13TIP • Do not forget the constant,C, when you write the antiderivative after evaluating an
indefinite integral, e.g.,∫
cosxdx=sinx+C.Test for Concavity and Points of Inflection
Test for Concavity
Letf be a differentiable function.- Iff′′>0 on an interval I, thenf is concave upward on I.
- Iff′′<0 on an interval I, thenf is concave downward on I.
(See Figures 8.2-14 and 8.2-15.)