5 Steps to a 5 AP Calculus BC 2019

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

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


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


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


3. If f′(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 7.2-10. Find the relative

extrema off.

Figure 7.2-10

Solution: (See Figure 7.2-11.)

–2 3

incr. decr. incr.

+– +

rel. max rel. min

x

f

f ′

Figure 7.2-11

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