3.4. The Second Derivative Test http://www.ck12.org
The example above had only one inflection point. But we can easily come up with examples of functions where
there is more than one point of inflection.Example 2:
Consider the functionf(x) =x^4 − 3 x^3 +x− 2.
We can see that the graph has two relative minimums, one relative maximum, and two inflection points (as indicated
by arrows).
In general we can use the following two tests for concavity and determining where we have relative maximums,
minimums, and inflection points.
Test for Concavity
Suppose thatIis some interval[a,b]in the domain offand thatfis continuous onI.
- Iff′′(x)>0 for allx∈I,then the graph offis concave upward onI.
- Iff′′(x)<0 for allx∈I,then the graph offis concave downward onI.
A consequence of this concavity test is the following test to identify extreme values off.
Second Derivative Test for Extrema
Suppose thatfis a continuous function nearcand thatcis a critical value off.Then
- Iff′′(c)> 0 ,thenfhas a relative minimum atx=c.
- Iff′′(c)< 0 ,thenfhas a relative maximum atx=c.
- Iff′′(c) = 0 ,then the test is inconclusive andx=cmay be a point of inflection.
Recall the graphf(x) =x^3 .We observed thatx= 0 ,and that there was neither a maximum nor minimum. The
Second Derivative Test cautions us that this may be the case since atf′′( 0 ) =0 atx= 0.
So now we wish to use all that we have learned from the First and Second Derivative Tests to sketch graphs of
functions. The following table provides a summary of the tests and can be a useful guide in sketching graphs.