Notefirst that we havea relativemaximumat and a relativeminimumat The slopesof the
tangentlineschangefrom positivefor to negativefor and then backto positivefor
. Fromthis we exampleinfer the followingtheorem:
FirstDerivativeTest
Supposethat is a continuousfunctionand that is a criticalvalueof Then:
- If changesfrom positiveto negativeat then has a local maximumat
- If changesfrom negativeto positiveat then has a local minimumat
- If doesnot changesign at then has neithera local maximumnor minimumat
Proofof thesethreeconclusionsis left to the reader.
Example3:
Our previousexampleshoweda graphthat had botha localmaximumand minimum.Let’s reconsider
and observethe grapharound Whathappensto the first derivativenear this value?
Example4:
Let'sconsiderthe function and observethe grapharound Whathappens
to the first derivativenear this value?