112 STEP 4. Review the Knowledge You Need to Score High
- 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. - 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.
- If f′(c)=0 andf′′(c)<0, then f(c) is a relative maximum.
- If f′(c)=0 andf′′(c)>0, then f(c) is a relative minimum.
- 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.