I can integrate MAX(sin X,cos X) for an interval of interest in CAS and in home. When plotting MAX(sin X,cos X), the plot is fine, but it will not find extremums. Likewise, I am not having success in finding fmax(sin X,cos X) in home using "X" or CAS using "x".

What is the correct procedure/syntax?

Is it possible to find extremums of functions in Home/CAS/graphing apps that use MAX such as MAX(sin x, cos x) ?

The Function app currently relies on the differentiating functionality of the CAS for extrema finding. I might add (NOTE: as has been said before, I'm not speaking for HP: there are no guarantees towards future releases or improvements etc.) a numerical extrema finder for Function / Fcn / Plot / Extremum. The CAS doesn't currently like differentiating MAX.

One possible workaround is to define V1 in the Advanced Graphing app as 0.5*(X+Y+ABS(X-Y)) and then define F1 as V1(SIN(X),COS(X)). Then try using Function / Plot / Fcn / Extremum (the CAS is more likely to differentiate ABS). This workaround also helps with Fcn / Slope and Fcn / Tangent (the CAS is similarly used for differentiation for these features).