When does an inequality indicate a maximum or minimum value?

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In summary: Thank you for the clarification. I understand now that the inequality only states the boundedness of the values and does not provide information about the specific max or min values.
  • #1
Mr Davis 97
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If I write ##-1 \le\cos x \le 1##, we all clearly know what this means, that not only is ##\cos x## contained in this interval but that its max is 1 and its min is -1. However, what if I write ##-1 \le \cos n \le 1##, where ##n \in \mathbb{N}##. What does the inequality mean in this case? This inequality is still true, but it doesn't say anything about the max or the min. I guess my question is this, in what cases can we suppose that an inequality is saying something about a max or a min, and in what cases is the inequality simply stating that something is bounded above or below?
 
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  • #2
Mr Davis 97 said:
If I write ##-1 \le\cos x \le 1##, we all clearly know what this means, that not only is ##\cos x## contained in this interval but that its max is 1 and its min is -1.
But only because we know it from some other sources.
However, what if I write ##-1 \le \cos n \le 1##, where ##n \in \mathbb{N}##. What does the inequality mean in this case?
The same, that the values of ##\cos n \in [-1,1]##.
This inequality is still true, but it doesn't say anything about the max or the min.
Neither did the first.
I guess my question is this, in what cases can we suppose that an inequality is saying something about a max or a min,...
Never, unless we add information. I mean, to know that ##\cos \mathbb{N}## doesn't reach the boundaries requires the knowledge, that ##\pi ## is transcendental. Not a trivial result.
... and in what cases is the inequality simply stating that something is bounded above or below?
Always.

There are of course situations where we can conclude something about the boundaries, e.g. in linear optimization circumstances, or if the function has monotony properties. Even the knowledge, that the maximum is inside a set requires a compact set and a continuous function.
 
  • #3
fresh_42 said:
But only because we know it from some other sources.

The same, that the values of ##\cos n \in [-1,1]##.

Neither did the first.

Never, unless we add information. I mean, to know that ##\cos \mathbb{N}## doesn't reach the boundaries requires the knowledge, that ##\pi ## is transcendental. Not a trivial result.

Always.

There are of course situations where we can conclude something about the boundaries, e.g. in linear optimization circumstances, or if the function has monotony properties. Even the knowledge, that the maximum is inside a set requires a compact set and a continuous function.
Alright, I think I understand now. One more thing though, as an example. By AM-GM, we have that ##x+1/x \ge 2##. In trying to find the minimum value of ##f(x) = x+1/x## with ##x \ge 0##, how can how can we conclude that 2 is the minimum, if technically all it is is a lower bound?
 
  • #4
Mr Davis 97 said:
Alright, I think I understand now. One more thing though, as an example. By AM-GM, we have that ##x+1/x \ge 2##. In trying to find the minimum value of ##f(x) = x+1/x## with ##x \ge 0##, how can how can we conclude that 2 is the minimum, if technically all it is is a lower bound?
Better to define ##x > 0##. You can either use the knowledge that ##f(1)=2## as you did, when you said ##\cos x## takes the boundary values ##\pm 1## because you already knew ##\cos 0 = 1## and ##\cos \pi = -1##, or by solving ##f\,'(x)=0## and look for the minima and maxima of this differential function. Or you prove that ##f## is strictly monotone decreasing on ##0<x\le 2## and strictly monotone increasing for ##x \geq 2 ##, which appears to be more work to do.
 
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  • #5
Mr Davis 97 said:
If I write ##-1 \le\cos x \le 1##, we all clearly know what this means, that not only is ##\cos x## contained in this interval but that its max is 1 and its min is -1. However, what if I write ##-1 \le \cos n \le 1##, where ##n \in \mathbb{N}##. What does the inequality mean in this case? This inequality is still true, but it doesn't say anything about the max or the min. I guess my question is this, in what cases can we suppose that an inequality is saying something about a max or a min, and in what cases is the inequality simply stating that something is bounded above or below?

We can also write ##-100 \leq \cos x \leq 100## and this is true as well. Really no problem here, but your statement contained more information.
 
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  • #6
In a closed bounded set, a continuous function always attains its extremal values (the general result is much more..well..general). This is fine to talk about for subsets of ##\mathbb R ##, but not for ##\mathbb N ##. Continuity is a topological property, which subsets do you call open sets in ##\mathbb N ##?

Naturally, ##\cos 0 ## is the maximum value, but you won't find any other multiple of ##\pi## in ##\mathbb N_0 ##. You can ask whether you get a minimum value too for some ## \mathbb N##, but that is not obvious.
 
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  • #7
nuuskur said:
for some ##\mathbb N##
bah.. for some ##n\in\mathbb N ## is what I meant. :wideeyed:
 
  • #8
Mr Davis 97 said:
If I write ##-1 \le\cos x \le 1##, we all clearly know what this means, that not only is ##\cos x## contained in this interval but that its max is 1 and its min is -1.
Not true. Just to focus on an important part of other comments, you are reading too much into ##-1 \le\cos x \le 1##. It says nothing about the min and max except that they must be somewhere in the closed interval [-1,1].
 
  • #9
FactChecker said:
Not true. Just to focus on an important part of other comments, you are reading too much into ##-1 \le\cos x \le 1##. It says nothing about the min and max except that they must be somewhere in the closed interval [-1,1].
To be pickier, by itself it does not guarantee that either min or max even exist. Although, as @nuuskur points out, the continutity of cos x does allow one to deduce existence.
 
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Related to When does an inequality indicate a maximum or minimum value?

1. What are inequalities?

Inequalities are mathematical expressions that compare two quantities or values. They are used to show the relationship between two numbers or variables, where one is greater than, less than, or not equal to the other.

2. Why are inequalities important?

Inequalities are important because they help us understand and solve real-world problems. They are also used in many areas of science, such as economics, physics, and biology, to describe relationships between different variables.

3. How do you solve inequalities?

To solve inequalities, you need to follow the same rules as solving equations, but with some additional steps. The goal is to isolate the variable on one side of the inequality sign. You can use inverse operations, such as addition, subtraction, multiplication, or division, to manipulate the inequality and find the solution.

4. What is the difference between solving equations and solving inequalities?

The main difference between solving equations and solving inequalities is that in equations, the goal is to find the exact value of the variable, while in inequalities, the goal is to find a range of values that satisfy the inequality. Also, when multiplying or dividing by a negative number in inequalities, you need to switch the direction of the inequality sign.

5. Are there any shortcuts or tricks to solving inequalities?

Yes, there are some shortcuts or rules that can help you solve inequalities more efficiently. For example, when multiplying or dividing both sides of an inequality by a positive number, you do not need to switch the direction of the inequality sign. Also, when adding or subtracting a positive number, the direction of the inequality remains the same, but when adding or subtracting a negative number, the direction of the inequality sign switches.

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