# For each real number x, let f(x) be the minimum of the numbers 4x+1,

For each real number x, let f(x) be the minimum of the numbers 4x+1, x+2, and -2x+4. What is the maximum value of f(x)?

Mark44
Mentor

For each real number x, let f(x) be the minimum of the numbers 4x+1, x+2, and -2x+4. What is the maximum value of f(x)?
Is this a homework problem?

Is this a homework problem?

No, Sir.
It is too taken from The Advanced Mathematical Thinking where the author said it can be solved by reversal tactic.

Sorry if it is incomplete since I copied whatever written in the book.

Bacle2

Is there a range for these numbers, i.e., is x any real number, integer, subset of these

or other?

Is there a range for these numbers, i.e., is x any real number, integer, subset of these

or other?

http://img214.imageshack.us/img214/5270/35240583.jpg [Broken]
http://img850.imageshack.us/img850/1995/68878748.jpg [Broken]

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Bacle2

Well, the reason I was asking is that , if the minimum of these numbers is done

over different subsets of the line, then the results will be different. The solution

is straightforward: compare the functions and see which of the three dominates

over which part of the domain, and construct a piecewise function: set

f1>f2 ,f1>f3 , f2>f3 , etc., and select, for each interval the smallest.