CK12 Calculus - Single Variable

The functionindicatedhere is strictlyincreasingon and and strictlydecreasingon and

We can now statethe theoremsthat relatederivativesof functionsto the increasing/decreasingproperties
of functions.

Theorem: If is continuouson interval then:

1. If for every then is strictlyincreasingin


2. If for every then is strictlydecreasingin
   Proof: We will provethe first statement.A similarmethodcan be usedto provethe secondstatementand
   is left as an exerciseto the student.

Consider with By the MeanValue Theorem,thereexists suchthat

By assumption, for every ; hence Also,note that
Hence

and
We can observethe consequencesof this theoremby observingthe tangentlinesof the followinggraph.
Notethe tangentlines to the graph,one in eachof the intervals