http://www.ck12.org Chapter 3. Applications of Derivatives
Note first that we have a relative maximum atx=aand a relative minimum atx=b.The slopes of the tangent lines
change from positive forx∈( 0 ,a)to negative forx∈(a,b)and then back to positive forx∈(b,+∞). From this we
example infer the following theorem:
First Derivative Test
Suppose thatfis a continuous function and thatx=cis a critical value off.Then:
- Iff′changes from positive to negative atx=c,thenfhas a local maximum atx=c.
- Iff′changes from negative to positive atx=c,thenfhas a local minimum atx=c.
- Iff′does not change sign atx=c,thenfhas neither a local maximum nor minimum atx=c.
Proof of these three conclusions is left to the reader.
Example 3:
Our previous example showed a graph that had both a local maximum and minimum. Let’s reconsiderf(x) =x^3 and
observe the graph aroundx= 0 .What happens to the first derivative near this value?
We observe that the tangent lines to the graph are positive on both sides ofx=0. The first derivative test (f′(x) = 3 x^2 )
verifies this fact, and that the slopes of the tangent line are positive for all nonzerox. Althoughf′( 0 ) =0, and sof
has a critical value atx=0, the third part of the First Derivative Test tells us that the failure off′to change sign at
x=0 means thatfhas neither a local minimum nor a local maximum atx=0.
Example 4:
Let’s consider the function f(x) =x^2 + 6 x−9 and observe the graph aroundx=− 3 .What happens to the first
derivative near this value?