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Find max of a Find root function


Can someone please assist me with the following sample problem.  Bottom line is that I'd like to use the FindRoot function to numerically solve an equation, that involves a parameter.  I'd then like to use the Findmax function that would yield the largest root for a specified range of the parameter.  Any help would be appreciated.  Thanks

Dan
I first define the function
f[a_?NumberQ]:=x/.Last[FindRoot[xSCos[a x],{x,.4}]]

and it evaluates nicely.

f[3]
0.39004

It even graphs nicely.

Plot[f[x],{x,2,3}]
[cid:image002.png at 01CB8019.CA736910]

But it doesn't like me asking for a max...


FindMaximum[{f[a],2=A3a=A33},{a,3}]

The errors I get are the following

FindRoot::nlnum :  The function value {1. -1. Cos[1. a]} is not a list of numbers with dimensions {1} at {x} = {1.}. ?
ReplaceAll::reps :  {FindRoot[xSCos[a x],{x,1}]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. ?
FindRoot::nlnum :  The function value {1. -1. Cos[1. a]} is not a list of numbers with dimensions {1} at {x} = {1.}. ?
ReplaceAll::reps :  {FindRoot[xSCos[a x],{x,1}]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. ?
FindRoot::nlnum :  The function value {1. -1. Cos[1. a]} is not a list of numbers with dimensions {1} at {x} = {1.}. ?
General::stop :  Further output of FindRoot::nlnum will be suppressed during this calculation. ?
ReplaceAll::reps :  {FindRoot[xSCos[a x],{x,1}]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. ?
General::stop :  Further output of ReplaceAll::reps will be suppressed during this calculation. ?
{-0.887727,{a=AE3.}}


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