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

In summary, the question asks for the maximum value of the function f(x), where for each real number x, f(x) is the minimum of the numbers 4x+1, x+2, and -2x+4. The solution involves comparing the three functions and constructing a piecewise function to find the maximum value of f(x). The range of x is not specified, so the solution may vary depending on the subset of the real numbers that is being considered. This problem is taken from the book "Advanced Mathematical Thinking" and the solution involves a reversal tactic.
  • #1
azizlwl
1,066
10
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)?
 
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  • #2


azizlwl said:
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?
 
  • #3


Mark44 said:
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.
 
  • #4


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

or other?
 
  • #5


Bacle2 said:
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
2.https://www.physicsforums.com/showthread.php?t=632495
http://img850.imageshack.us/img850/1995/68878748.jpg
 
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  • #6


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.
 

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

1. What is the function f(x)?

The function f(x) is defined as the minimum value among the set of numbers 4x+1, for any real number x.

2. How do you find the minimum value of f(x)?

To find the minimum value of f(x), you simply plug in different values of x into the expression 4x+1 and determine the smallest result.

3. Can f(x) have a maximum value?

No, f(x) cannot have a maximum value as it is defined as the minimum value among a set of numbers. However, it can approach infinity as x approaches negative infinity.

4. What is the domain of f(x)?

The domain of f(x) is all real numbers, as there are no restrictions on the values of x in the expression 4x+1.

5. Is f(x) a continuous function?

Yes, f(x) is a continuous function as it is defined for all real numbers and there are no breaks or gaps in its graph.

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