5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 17:21

146 STEP 4. Review the Knowledge You Need to Score High

1. Iff′(x) changes from positive to negative atx=c(f′>0 forx<cand f′<0 for
   x>c), then fhas a relative maximum atc.


2. Iff′(x) changes from negative to positive atx=c(f′<0 forx<cand f′>0 for
   x>c), then fhas a relative minimum atc.

Second Derivative Test for Relative Extrema

Letf be a continuous function at a numberc.

1. Iff′(c)=0 andf′′(c)<0, then f(c) is a relative maximum.


2. Iff′(c)=0 andf′′(c)>0, then f(c) is a relative minimum.


3. Iff′(c)=0 andf′′(c) =0, then the test is inconclusive. Use the First Derivative Test.

Example 1

The graph off′, the derivative of a functionf, is shown in Figure 8.2-10. Find the relative

extrema off.

y

f′

x

• 2 0 3

Figure 8.2-10

Solution:(See Figure 8.2-11.)

• 23
  incr. decr. incr.

+ – +

rel. max rel. min

x

f

f′

Figure 8.2-11

Thus,fhas a relative maximum atx=−2, and a relative minimum atx=3.