MHB No Max for Functions: Examples & Explanations

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Hello! (Wave)

If $f: \mathbb{R}^n \to \mathbb{R} (n \in \mathbb{N})$ and $D \subset \mathbb{R}^n$ then it doesn't generally hold that there exists a $\max_{x \in D} f(x)$, i.e. that there is a $\overline{x} \in D$ such that $f(\overline{x}) \geq f(x)$ for each $x \in D$.

For example if $n=1$ and $f(x)=x$, $D=(0,1)$ then $f$ doesn't have a maximum in $(0,1)$. Give examples for $n \geq 2$.

Hint: $f(x_1,x_2)=x_1+x_2$ and find a space $D \subset \mathbb{R}^2$ so that $f$ has't maximum in $D$.How can we find a $D$ such that $f(x_1,x_2)=x_1+x_2$ doesn't have a maximum in $D$ ?Would it be right as follows? $$$$

We consider $D=((1,1), (2,2))$.

If the maximum is in $((1,1),(2,2))$, it holds that $\max_{x \in D} f(x)\subset (2,2)$, say it is $(2- \epsilon_1, 2-\epsilon_2)$. From the density of rationals, we have that there are $q_1, q_2 \in \mathbb{Q}$ such that $2- \epsilon_1< q_1< 2$ and $2- \epsilon_2< q_2< 2 $.

But then $(q_1, q_2 ) \in ((1,1), (2,2))$ and $ (2- \epsilon_1, 2-\epsilon_2) \subset (q_1, q_2)$. Thus $(2- \epsilon_1, 2-\epsilon_2)$ cannot be the maximum.

$$$$

In general, can we find an open and bounded interval $D$ such that a function $f$ has a maximum in it?
 
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evinda said:
Hello! (Wave)

If $f: \mathbb{R}^n \to \mathbb{R} (n \in \mathbb{N})$ and $D \subset \mathbb{R}^n$ then it doesn't generally hold that there exists a $\max_{x \in D} f(x)$, i.e. that there is a $\overline{x} \in D$ such that $f(\overline{x}) \geq f(x)$ for each $x \in D$.

For example if $n=1$ and $f(x)=x$, $D=(0,1)$ then $f$ doesn't have a maximum in $(0,1)$. Give examples for $n \geq 2$.

Hint: $f(x_1,x_2)=x_1+x_2$ and find a space $D \subset \mathbb{R}^2$ so that $f$ has't maximum in $D$.How can we find a $D$ such that $f(x_1,x_2)=x_1+x_2$ doesn't have a maximum in $D$ ?

Hi evinda! (Mmm)

Suppose we pick $D$ to be an open ball.
How would that work out? (Wondering)
Would it be right as follows? $$$$

We consider $D=((1,1), (2,2))$.

If the maximum is in $((1,1),(2,2))$, it holds that $\max_{x \in D} f(x)\subset (2,2)$

Wouldn't the maximum be some value in $\mathbb R$ instead of a point (or a subset)? (Wondering)

For the $D$ you suggest, we would only have 2 values for $f$.
Let's call them $f_1 = f(1,1)$ and $f_2 = f(2,2)$.
Whichever of them matches the maximum, the corresponding $\overline x$ will be in $D$. (Worried)
 
I like Serena said:
Suppose we pick $D$ to be an open ball.
How would that work out? (Wondering)

What open ball could we pick for example? (Thinking)
I like Serena said:
Wouldn't the maximum be some value in $\mathbb R$ instead of a point (or a subset)? (Wondering)

For the $D$ you suggest, we would only have 2 values for $f$.
Let's call them $f_1 = f(1,1)$ and $f_2 = f(2,2)$.
Whichever of them matches the maximum, the corresponding $\overline x$ will be in $D$. (Worried)
So could we compute $f_1 = f(1,1)$ and $f_2 = f(2,2)$ although the intervals of the space $D=((1,1), (2,2))$ are open?
 
evinda said:
What open ball could we pick for example? (Thinking)

Say the open unit ball around the origin:
$$D=\{(x,y) : x^2+y^2 < 1 \}$$

So could we compute $f_1 = f(1,1)$ and $f_2 = f(2,2)$ although the intervals of the space $D=((1,1), (2,2))$ are open?

Perhaps I misunderstood.
What did you mean by $D=((1,1), (2,2))$? (Wondering)
If it represents the straight line between the points $(1,1)$ and $(2,2)$ without its endpoints, that would also do just fine. (Nod)
Then the range of the given $f$ would be the interval $(2,4)$, so its maximum would be $4$, except that is not included in the range.
 
I like Serena said:
Say the open unit ball around the origin:
$$D=\{(x,y) : x^2+y^2 < 1 \}$$

How do we know that it is an open ball? (Thinking)

I like Serena said:
Perhaps I misunderstood.
What did you mean by $D=((1,1), (2,2))$? (Wondering)
If it represents the straight line between the points $(1,1)$ and $(2,2)$ without its endpoints, that would also do just fine. (Nod)
Then the range of the given $f$ would be the interval $(2,4)$, so its maximum would be $4$, except that is not included in the range.

I don't know.. I am a little confused right now. Should we find a space $D \subset \mathbb{R}^2$ of the form $D=((x_1, x_2))$ ? (Sweating)
 
evinda said:
How do we know that it is an open ball? (Thinking)

It's the definition of an open ball. Quoting you in your other thread:
evinda said:
$B_d(x, \epsilon) := \{ y \in \mathbb{R}^n: d(x,y)< \epsilon\}$ -> open ball with center $x$ and radius $\epsilon$.

To see that the ball is indeed open, we can check if we can find a mini-ball around each point in the ball that is completely inside the ball... (Thinking)

I don't know.. I am a little confused right now. Should we find a space $D \subset \mathbb{R}^2$ of the form $D=((x_1, x_2))$ ? (Sweating)

Let's start with your notation $D=((x_1, x_2))$... what does it mean? (Wondering)
 
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I like Serena said:
It's the definition of an open ball. Quoting you in your other thread:To see that it the ball is indeed open, we can check if we can find a mini-ball around each point in the ball that is completely inside the ball... (Thinking)

How can we determine if we can find a mini-ball around each point in the ball that is completely inside the ball?

Which points $(x,y)$ do we check to see if they satisfy the relation $x^2+y^2<1$ , picking $D=\{ (x,y): x^2+y^2<1\} $? (Thinking)

I like Serena said:
Let's start with your notation $D=((x_1, x_2))$... what does it mean? (Wondering)

That the values that the function $f$ can take belong to the interval $(x_1,x_2)$ ? Or am I wrong? (Thinking)
 
evinda said:
How can we determine if we can find a mini-ball around each point in the ball that is completely inside the ball?

Which points $(x,y)$ do we check to see if they satisfy the relation $x^2+y^2<1$ , picking $D=\{ (x,y): x^2+y^2<1\} $? (Thinking)

Let's pick a point $(x_1,y_1)$ that is an element of the ball $D$.
That is, it satisfies $x_1^2+y_1^2<1$.
Now the question is, can we find an $\epsilon > 0$ so small that the disk around the point $(x_1,y_1)$ with radius $\epsilon$ fits completely inside $D$?
Which $\epsilon$ could be pick to achieve that? (Wondering)

That the values that the function $f$ can take belong to the interval $(x_1,x_2)$ ? Or am I wrong? (Thinking)

I don't think so. (Worried)

We're running into the ambiguity what $(x_1,x_2)$ means. Is it a point in $\mathbb R^2$ or is it an open interval from $x_1$ to $x_2$?
When this happens, we often need to deduce from the context what was intended.
In this case $D$ is a subset of $\mathbb R^2$, so I'd interpret $(x_1,x_2)$ as a point unless it is specifically mentioned that it is an interval. Moreover, from your opening post we can also deduce that $(x_1,x_2)$ must be a point.
Ideally, we always state explicitly what is intended. (Nerd)

Either way, the extra set of parentheses in $D = ((x_1,x_2))$ has no known meaning to me.
I can only guess what it might mean. (Doh)
 
I like Serena said:
Let's pick a point $(x_1,y_1)$ that is an element of the ball $D$.
That is, it satisfies $x_1^2+y_1^2<1$.
Now the question is, can we find an $\epsilon > 0$ so small that the disk around the point $(x_1,y_1)$ with radius $\epsilon$ fits completely inside $D$?
Which $\epsilon$ could be pick to achieve that? (Wondering)

So do we have to check if there is an $\epsilon>0$ such that $(x_1- \epsilon)^2+(y_1-\epsilon)^2<1$?

So if we pick $D=\{ (x,y): x^2+y^2<1 \}$ in order to show that $f(x_1,x_2)=x_1+x_2$ doesn't have a maximum in $D$ how can we proceed?
Can we assume that $f$ has a maximum in $D$ at the point $(\overline{x_1}, \overline{x_2})$?
If so, then $\overline{x_1}+\overline{x_2}$ is the greatest value for which $\overline{x_1}+\overline{x_2}<1$.
How could we continue?
 
  • #10
evinda said:
So do we have to check if there is an $\epsilon>0$ such that $(x_1- \epsilon)^2+(y_1-\epsilon)^2<1$?

Yup. (Thinking)

So if we pick $D=\{ (x,y): x^2+y^2<1 \}$ in order to show that $f(x_1,x_2)=x_1+x_2$ doesn't have a maximum in $D$ how can we proceed?
Can we assume that $f$ has a maximum in $D$ at the point $(\overline{x_1}, \overline{x_2})$?
If so, then $\overline{x_1}+\overline{x_2}$ is the greatest value for which $\overline{x_1}+\overline{x_2}<1$.
How could we continue?

The maximum on the closure of $D$ is at $P = (\frac 12 \sqrt 2, \frac 12 \sqrt 2)$ and has the value $\sqrt 2$.
However, that point is on the boundary and does not belong to $D$ itself.
Any point in $D$ will always have a value that is slightly less, and whichever point we pick, we can always pick a point closer to $P$, which has a higher value.
In other words, $D$ does not have a maximum.
It does have a supremum, which is $\sqrt 2$.
 
  • #11
I like Serena said:
Yup. (Thinking)

Could we pick for example $\epsilon=x_1$ ? (Thinking)

I like Serena said:
The maximum on the closure of $D$ is at $P = (\frac 12 \sqrt 2, \frac 12 \sqrt 2)$ and has the value $\sqrt 2$.
How do we deduce this?

I like Serena said:
Any point in $D$ will always have a value that is slightly less, and whichever point we pick, we can always pick a point closer to $P$, which has a higher value.

Could you explain it further to me?
 
  • #12
evinda said:
Could we pick for example $\epsilon=x_1$ ? (Thinking)

Not generally.
Suppose $(x_1,y_1) = (0.9, 0)$, then if we pick $\epsilon=0.9$ the resulting disk won't fit inside $D$. (Worried)
How do we deduce this?

Could you explain it further to me?

How would you find the maximum of a regular function, say $f: x \mapsto x^2$, on the interval $[-1,2]$? (Wondering)
 
  • #13
I like Serena said:
Not generally.
Suppose $(x_1,y_1) = (0.9, 0)$, then if we pick $\epsilon=0.9$ the resulting disk won't fit inside $D$. (Worried)

So can't we find an $\epsilon$ for the general case? (Thinking)

I like Serena said:
How would you find the maximum of a regular function, say $f: x \mapsto x^2$, on the interval $[-1,2]$? (Wondering)

We find that $f$ is decreasing for $x<0$ and increasing for $x>0$, so it achieves its maximum for $x=2$.
 
  • #14
evinda said:
So can't we find an $\epsilon$ for the general case? (Thinking)

What is the distance of $(x_1, y_1)$ to the edge of $D$? (Wondering)
Perhaps we can use that as $\epsilon$.
We find that $f$ is decreasing for $x<0$ and increasing for $x>0$, so it achieves its maximum for $x=2$.

Yep!
And what is the maximum if the interval is $(-1,2)$? (Wondering)
 
  • #15
I like Serena said:
What is the distance of $(x_1, y_1)$ to the edge of $D$? (Wondering)
Perhaps we can use that as $\epsilon$.

With edge of $D$ do you mean the circle with center $(0,0)$ and radius $1$?
I like Serena said:
Yep!
And what is the maximum if the interval is $(-1,2)$? (Wondering)

There is no maximum, only a supremum at the point $x=2$.
So does it hold that any function does not have a maximum on an open and bounded interval, but just a supremum? (Thinking)
 
  • #16
evinda said:
With edge of $D$ do you mean the circle with center $(0,0)$ and radius $1$?

Yes. (Sweating)
There is no maximum, only a supremum at the point $x=2$.
So does it hold that any function does not have a maximum on an open and bounded interval, but just a supremum? (Thinking)

How about $f: x \mapsto 2-x^2$. Does it have a maximum on $(-1,2)$? (Wondering)
 
  • #17
I like Serena said:
Yes. (Sweating)

How can we find the distance? (Sweating)
I like Serena said:
How about $f: x \mapsto 2-x^2$. Does it have a maximum on $(-1,2)$? (Wondering)

No, $f$ has just a supremum at the point $x=-1$.
 
  • #18
evinda said:
How can we find the distance? (Sweating)

The distance of $(x_1,y_1)$ to the origin is $\sqrt{x_1^2+y_1^2}$.
The distance of the origin to any point on the boundary of $D$ is $1$. (It's a unit disk.)
Therefore the distance of $(x_1,y_1)$ to the boundary is $1-\sqrt{x_1^2+y_1^2}$.
If we pick that as $\epsilon$ we're good to go.
No, $f$ has just a supremum at the point $x=-1$.

Let's see... (Thinking)

$f(-1)=2-(-1)^2=1$, but $f(0)=2-0^2=2$. Isn't that bigger? (Wondering)
 
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  • #19
I like Serena said:
The distance of $(x_1,y_1)$ to the origin is $\sqrt{x_1^2+y_1^2}$.
The distance of the origin to any point on the boundary of $D$ is $1$. (It's a unit disk.)
Therefore the distance of $(x_1,y_1)$ to the boundary is $1-\sqrt{x_1^2+y_1^2}$.
If we pick that as $\epsilon$ we're good to go.

I haven't really understood why the distance of $(x_1,y_1)$ to the boundary is $1-\sqrt{x_1^2+y_1^2}$. (Sweating)
Could you explain it further to me?

I like Serena said:
Let's see... (Thinking)

$f(-1)=2-(-1)^2=1$, but $f(0)=2-0^2=2$. Isn't that bigger? (Wondering)

Yes, so $f$ achieves its maximum at the point $x=0$.
 
  • #20
evinda said:
I haven't really understood why the distance of $(x_1,y_1)$ to the boundary is $1-\sqrt{x_1^2+y_1^2}$. (Sweating)
Could you explain it further to me?

Let's consider the real line and suppose we have some $0\le x<1$.
The distance from $0$ to $x$ is $|x|$, while the distance from $x$ to $1$ is $1-|x|$. Yes? (Thinking)

The same holds in any number of dimensions: the distance from $\mathbf 0$ to $\mathbf x$ is $||\mathbf x||$, while the distance from $\mathbf x$ to the boundary of the unit ball is $1-||\mathbf x||$. (Thinking)
Yes, so $f$ achieves its maximum at the point $x=0$.

Aha! So not all functions have their maximum or supremum on the boundary! (Smirk)
 
  • #21
I like Serena said:
Let's consider the real line and suppose we have some $0\le x<1$.
The distance from $0$ to $x$ is $|x|$, while the distance from $x$ to $1$ is $1-|x|$. Yes? (Thinking)

The same holds in any number of dimensions: the distance from $\mathbf 0$ to $\mathbf x$ is $||\mathbf x||$, while the distance from $\mathbf x$ to the boundary of the unit ball is $1-||\mathbf x||$. (Thinking)

A ok. And in our case do we take the euclidean norm? (Thinking)

I like Serena said:
Aha! So not all functions have their maximum or supremum on the boundary! (Smirk)

But how do we deduce in our case when we have the function $f(x_1,x_2)=x_1+x_2$ and $D=\{ (x,y): x^2+y^2<1\}$ that the maximum is on the boundary? Because of the fact that the fuction is increasing? (Thinking)
 
  • #22
evinda said:
A ok. And in our case do we take the euclidean norm? (Thinking)

Yes.

But how do we deduce in our case when we have the function $f(x_1,x_2)=x_1+x_2$ and $D=\{ (x,y): x^2+y^2<1\}$ that the maximum is on the boundary? Because of the fact that the fuction is increasing? (Thinking)

It's a linear function.
Linear functions always have their maximum at the boundary. (Nerd)

To find where, we can go 2 ways:
1. Parametrize the boundary, substitute, and find the maximum.
2. Use symmetry considerations to find the maximum.

For approach 1 that means substituting $x=\cos\phi, y=\sin\phi$ in $f(x,y)$, take the derivative with respect to $\phi$, set it to zero, and solve. (Thinking)

For approach 2 that means that we consider that f(x,y) is symmetric in x and y. (Thinking)
 
  • #23
I like Serena said:
Yes.
It's a linear function.
Linear functions always have their maximum at the boundary. (Nerd)

To find where, we can go 2 ways:
1. Parametrize the boundary, substitute, and find the maximum.

For approach 1 that means substituting $x=\cos\phi, y=\sin\phi$ in $f(x,y)$, take the derivative with respect to $\phi$, set it to zero, and solve. (Thinking)
So is it like that?

$$f(x,y)=f(\phi)= \cos \phi + \sin \phi$$

$$f'(\phi)=-\sin \phi+ \cos \phi$$

$$f'(\phi)=0 \Rightarrow \sin \phi= \cos \phi \Rightarrow \phi=\frac{\pi}{4} \text{ or } \phi=-\frac{3 \pi }{4}$$
So, $x=y= \frac{\sqrt{2}}{2}$.

Is it a general methodoly to find the maximum of a linear function?
 
  • #24
evinda said:
So is it like that?

$$f(x,y)=f(\phi)= \cos \phi + \sin \phi$$

$$f'(\phi)=-\sin \phi+ \cos \phi$$

$$f'(\phi)=0 \Rightarrow \sin \phi= \cos \phi \Rightarrow \phi=\frac{\pi}{4} \text{ or } \phi=-\frac{3 \pi }{4}$$
So, $x=y= \frac{\sqrt{2}}{2}$.

Is it a general methodoly to find the maximum of a linear function?

Good! (Smile)

It's a general methodology to find the maximum of any function on a boundary.

For a linear function we can make it easier for ourselves if we pick a boundary that consists of line segments, such as the unit square. Then the maximum must be on a corner. (Mmm)
 
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