If Rolle's Theorem is assumed to be true, doesn't that prove the MVT?
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If we assume that Rolle's Theorem is true is it practical to say that it also proves the MVT?
My reasoning is that even though Rolle's Theorem is the special case for when $f(a)=f(b)$ and the secant line between $(a,f(a))$ and $(b,f(b))$ is horizontal, doesn't that mean you can 'rotate' and 'stretch' the function $f$ and thus 'stretch' and 'rotate' the straight line between $(a,f(a))$ and $(b,f(b))$ as well as the line tangent to $(c,f(c))$ to maintain its symmetry to the line secant between $(a,f(a))$ and $(b,f(b))$ to thus prove that for the straight line intersecting points $(a,f(a))$ and $(b,f(b))$, there must be a point $c$ on $[a,b]$ for which $f'(c)$ is parallel to the aforementioned line between $(a,f(a))$ and $(b, f(b))$?
I feel that this could be better explained with something like linear algebra. I do not know it well enough to use it.
EDIT: My question was just as unclear as I thought it was. I am going to try and explain it differently:
For a function $f$, if it is differentiable/continuous over $[a,b] $/$(a,b)$ etc. then there is a straight line between $(a, f(a))$ and $(b,f(b))$. You can rotate $f$ around the origin until this straight line is parallel to the x-axis. By Rolle's Theorem, there must exist a $c$ on the transformed $(a,b)$ such that $f'(c)=0$ and thus there is at least one line tangent to $f$ on $(a,b)$ that is parallel to the secant of $(a,f(a))$ and $(b,f(b))$. Then, you can 'undo' the original rotation of $f$ and know that the lines proven to exist by Rolle's Theorem exist with the same symmetries with respect to $f$, meaning the MVT must hold even when the secant line is not horizontal.
calculus linear-algebra linear-transformations rotations rolles-theorem
 |Â
show 8 more comments
up vote
4
down vote
favorite
If we assume that Rolle's Theorem is true is it practical to say that it also proves the MVT?
My reasoning is that even though Rolle's Theorem is the special case for when $f(a)=f(b)$ and the secant line between $(a,f(a))$ and $(b,f(b))$ is horizontal, doesn't that mean you can 'rotate' and 'stretch' the function $f$ and thus 'stretch' and 'rotate' the straight line between $(a,f(a))$ and $(b,f(b))$ as well as the line tangent to $(c,f(c))$ to maintain its symmetry to the line secant between $(a,f(a))$ and $(b,f(b))$ to thus prove that for the straight line intersecting points $(a,f(a))$ and $(b,f(b))$, there must be a point $c$ on $[a,b]$ for which $f'(c)$ is parallel to the aforementioned line between $(a,f(a))$ and $(b, f(b))$?
I feel that this could be better explained with something like linear algebra. I do not know it well enough to use it.
EDIT: My question was just as unclear as I thought it was. I am going to try and explain it differently:
For a function $f$, if it is differentiable/continuous over $[a,b] $/$(a,b)$ etc. then there is a straight line between $(a, f(a))$ and $(b,f(b))$. You can rotate $f$ around the origin until this straight line is parallel to the x-axis. By Rolle's Theorem, there must exist a $c$ on the transformed $(a,b)$ such that $f'(c)=0$ and thus there is at least one line tangent to $f$ on $(a,b)$ that is parallel to the secant of $(a,f(a))$ and $(b,f(b))$. Then, you can 'undo' the original rotation of $f$ and know that the lines proven to exist by Rolle's Theorem exist with the same symmetries with respect to $f$, meaning the MVT must hold even when the secant line is not horizontal.
calculus linear-algebra linear-transformations rotations rolles-theorem
2
If the question is "Can the mean value theorem be proved using Rolle's theorem?" the answer is "Yes." However, I don't see what this has to do with linear algebra, and I am very confused by your exposition.
â Xander Henderson
Oct 1 at 4:06
1
Did you read any notes or books on a proof of MVT? It should be clear that Rolle's theorem is used in any proof. If it is not clear to you, I can write an outline for you.
â edm
Oct 1 at 4:07
3
This is indeed how the MVT is often proved.
â Bungo
Oct 1 at 4:07
Perhaps what you mean by "rotate and stretch" is shear mapping.
â edm
Oct 1 at 4:15
2
@Matt There's a problem with your idea of rotating: it may cause the rotated graph to no longer be a function!
â Theo Bendit
Oct 1 at 4:40
 |Â
show 8 more comments
up vote
4
down vote
favorite
up vote
4
down vote
favorite
If we assume that Rolle's Theorem is true is it practical to say that it also proves the MVT?
My reasoning is that even though Rolle's Theorem is the special case for when $f(a)=f(b)$ and the secant line between $(a,f(a))$ and $(b,f(b))$ is horizontal, doesn't that mean you can 'rotate' and 'stretch' the function $f$ and thus 'stretch' and 'rotate' the straight line between $(a,f(a))$ and $(b,f(b))$ as well as the line tangent to $(c,f(c))$ to maintain its symmetry to the line secant between $(a,f(a))$ and $(b,f(b))$ to thus prove that for the straight line intersecting points $(a,f(a))$ and $(b,f(b))$, there must be a point $c$ on $[a,b]$ for which $f'(c)$ is parallel to the aforementioned line between $(a,f(a))$ and $(b, f(b))$?
I feel that this could be better explained with something like linear algebra. I do not know it well enough to use it.
EDIT: My question was just as unclear as I thought it was. I am going to try and explain it differently:
For a function $f$, if it is differentiable/continuous over $[a,b] $/$(a,b)$ etc. then there is a straight line between $(a, f(a))$ and $(b,f(b))$. You can rotate $f$ around the origin until this straight line is parallel to the x-axis. By Rolle's Theorem, there must exist a $c$ on the transformed $(a,b)$ such that $f'(c)=0$ and thus there is at least one line tangent to $f$ on $(a,b)$ that is parallel to the secant of $(a,f(a))$ and $(b,f(b))$. Then, you can 'undo' the original rotation of $f$ and know that the lines proven to exist by Rolle's Theorem exist with the same symmetries with respect to $f$, meaning the MVT must hold even when the secant line is not horizontal.
calculus linear-algebra linear-transformations rotations rolles-theorem
If we assume that Rolle's Theorem is true is it practical to say that it also proves the MVT?
My reasoning is that even though Rolle's Theorem is the special case for when $f(a)=f(b)$ and the secant line between $(a,f(a))$ and $(b,f(b))$ is horizontal, doesn't that mean you can 'rotate' and 'stretch' the function $f$ and thus 'stretch' and 'rotate' the straight line between $(a,f(a))$ and $(b,f(b))$ as well as the line tangent to $(c,f(c))$ to maintain its symmetry to the line secant between $(a,f(a))$ and $(b,f(b))$ to thus prove that for the straight line intersecting points $(a,f(a))$ and $(b,f(b))$, there must be a point $c$ on $[a,b]$ for which $f'(c)$ is parallel to the aforementioned line between $(a,f(a))$ and $(b, f(b))$?
I feel that this could be better explained with something like linear algebra. I do not know it well enough to use it.
EDIT: My question was just as unclear as I thought it was. I am going to try and explain it differently:
For a function $f$, if it is differentiable/continuous over $[a,b] $/$(a,b)$ etc. then there is a straight line between $(a, f(a))$ and $(b,f(b))$. You can rotate $f$ around the origin until this straight line is parallel to the x-axis. By Rolle's Theorem, there must exist a $c$ on the transformed $(a,b)$ such that $f'(c)=0$ and thus there is at least one line tangent to $f$ on $(a,b)$ that is parallel to the secant of $(a,f(a))$ and $(b,f(b))$. Then, you can 'undo' the original rotation of $f$ and know that the lines proven to exist by Rolle's Theorem exist with the same symmetries with respect to $f$, meaning the MVT must hold even when the secant line is not horizontal.
calculus linear-algebra linear-transformations rotations rolles-theorem
calculus linear-algebra linear-transformations rotations rolles-theorem
edited Oct 1 at 10:29
asked Oct 1 at 3:57
nine-hundred
19010
19010
2
If the question is "Can the mean value theorem be proved using Rolle's theorem?" the answer is "Yes." However, I don't see what this has to do with linear algebra, and I am very confused by your exposition.
â Xander Henderson
Oct 1 at 4:06
1
Did you read any notes or books on a proof of MVT? It should be clear that Rolle's theorem is used in any proof. If it is not clear to you, I can write an outline for you.
â edm
Oct 1 at 4:07
3
This is indeed how the MVT is often proved.
â Bungo
Oct 1 at 4:07
Perhaps what you mean by "rotate and stretch" is shear mapping.
â edm
Oct 1 at 4:15
2
@Matt There's a problem with your idea of rotating: it may cause the rotated graph to no longer be a function!
â Theo Bendit
Oct 1 at 4:40
 |Â
show 8 more comments
2
If the question is "Can the mean value theorem be proved using Rolle's theorem?" the answer is "Yes." However, I don't see what this has to do with linear algebra, and I am very confused by your exposition.
â Xander Henderson
Oct 1 at 4:06
1
Did you read any notes or books on a proof of MVT? It should be clear that Rolle's theorem is used in any proof. If it is not clear to you, I can write an outline for you.
â edm
Oct 1 at 4:07
3
This is indeed how the MVT is often proved.
â Bungo
Oct 1 at 4:07
Perhaps what you mean by "rotate and stretch" is shear mapping.
â edm
Oct 1 at 4:15
2
@Matt There's a problem with your idea of rotating: it may cause the rotated graph to no longer be a function!
â Theo Bendit
Oct 1 at 4:40
2
2
If the question is "Can the mean value theorem be proved using Rolle's theorem?" the answer is "Yes." However, I don't see what this has to do with linear algebra, and I am very confused by your exposition.
â Xander Henderson
Oct 1 at 4:06
If the question is "Can the mean value theorem be proved using Rolle's theorem?" the answer is "Yes." However, I don't see what this has to do with linear algebra, and I am very confused by your exposition.
â Xander Henderson
Oct 1 at 4:06
1
1
Did you read any notes or books on a proof of MVT? It should be clear that Rolle's theorem is used in any proof. If it is not clear to you, I can write an outline for you.
â edm
Oct 1 at 4:07
Did you read any notes or books on a proof of MVT? It should be clear that Rolle's theorem is used in any proof. If it is not clear to you, I can write an outline for you.
â edm
Oct 1 at 4:07
3
3
This is indeed how the MVT is often proved.
â Bungo
Oct 1 at 4:07
This is indeed how the MVT is often proved.
â Bungo
Oct 1 at 4:07
Perhaps what you mean by "rotate and stretch" is shear mapping.
â edm
Oct 1 at 4:15
Perhaps what you mean by "rotate and stretch" is shear mapping.
â edm
Oct 1 at 4:15
2
2
@Matt There's a problem with your idea of rotating: it may cause the rotated graph to no longer be a function!
â Theo Bendit
Oct 1 at 4:40
@Matt There's a problem with your idea of rotating: it may cause the rotated graph to no longer be a function!
â Theo Bendit
Oct 1 at 4:40
 |Â
show 8 more comments
4 Answers
4
active
oldest
votes
up vote
7
down vote
accepted
A proof of MVT used Rolle's theorem explicitly.
Consider a continuous function $f:[a,b]toBbb R$ that is differentiable on $(a,b)$. We are to prove MVT, i.e. to prove there exists a point $cin(a,b)$ such that $f'(c)=fracf(b)-f(a)b-a$, assuming that Rolle's theorem is true.
Define a new function $g:[a,b]toBbb R$ by $g(x)=f(x)-fracf(b)-f(a)b-ax$. Verify yourself that $g$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Verify also that $g(a)=g(b)$. So we can apply Rolle's theorem to this function $g$, to obtain a point $cin(a,b)$ such that $g'(c)=0$. Note that the derivative of $g$ is $g'(x)=f'(x)-fracf(b)-f(a)b-a$. The point $c$ also satisfies $f'(c)-fracf(b)-f(a)b-a=0$.
add a comment |Â
up vote
4
down vote
You don't need any linear transformations in $mathbbR^2$; you can do this by subtracting a linear polynomial from your function (which is, itself, a shear map). Simply take your continuous function $f : [a, b] to mathbbR$ that is differentiable on $(a, b)$, and let $g(x) = f(x) - fracf(b) - f(a)b - a (x - a) - f(a)$. In particular $f(a) = f(b) = 0$, and when you apply Rolle's theorem to find $c in (a, b)$ such that $g'(c) = 0$, you'll find that
$$f'(c) = fracf(b)- f(a)b - a.$$
add a comment |Â
up vote
1
down vote
Since none of the existing answers have explicitly pointed out your error:
For a function f, if it is differentiable/continuous over [a,b]/(a,b) etc. then there is a straight line between (a,f(a)) and (b,f(b)). You can rotate f around the origin until this straight line is parallel to the x-axis. [...]
No you cannot do this! You can apply Rolle's theorem to any differentiable function, but rotating the graph of a function in the cartesian plane may not result in a graph of another function, much less a differentiable function. This is the reason edm mentioned a shear in the comments, because a shear with an invariant vertical line is indeed the kind of transformation that would preserve differentiable functions.
add a comment |Â
up vote
-1
down vote
Not really.
Rolle's theorum is a special case of Legrange's MVT. Not the other way around. So just by assuming Rolle's theorum, MVT is not automatically proved. We do of course use RT in the MVT proof. But just because Rolle's theorum is true, MVT need'nt be.
If you are wondering why we use this special case to prove the MVT, it is because Rolle's theorum appeared first.
New contributor
add a comment |Â
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
accepted
A proof of MVT used Rolle's theorem explicitly.
Consider a continuous function $f:[a,b]toBbb R$ that is differentiable on $(a,b)$. We are to prove MVT, i.e. to prove there exists a point $cin(a,b)$ such that $f'(c)=fracf(b)-f(a)b-a$, assuming that Rolle's theorem is true.
Define a new function $g:[a,b]toBbb R$ by $g(x)=f(x)-fracf(b)-f(a)b-ax$. Verify yourself that $g$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Verify also that $g(a)=g(b)$. So we can apply Rolle's theorem to this function $g$, to obtain a point $cin(a,b)$ such that $g'(c)=0$. Note that the derivative of $g$ is $g'(x)=f'(x)-fracf(b)-f(a)b-a$. The point $c$ also satisfies $f'(c)-fracf(b)-f(a)b-a=0$.
add a comment |Â
up vote
7
down vote
accepted
A proof of MVT used Rolle's theorem explicitly.
Consider a continuous function $f:[a,b]toBbb R$ that is differentiable on $(a,b)$. We are to prove MVT, i.e. to prove there exists a point $cin(a,b)$ such that $f'(c)=fracf(b)-f(a)b-a$, assuming that Rolle's theorem is true.
Define a new function $g:[a,b]toBbb R$ by $g(x)=f(x)-fracf(b)-f(a)b-ax$. Verify yourself that $g$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Verify also that $g(a)=g(b)$. So we can apply Rolle's theorem to this function $g$, to obtain a point $cin(a,b)$ such that $g'(c)=0$. Note that the derivative of $g$ is $g'(x)=f'(x)-fracf(b)-f(a)b-a$. The point $c$ also satisfies $f'(c)-fracf(b)-f(a)b-a=0$.
add a comment |Â
up vote
7
down vote
accepted
up vote
7
down vote
accepted
A proof of MVT used Rolle's theorem explicitly.
Consider a continuous function $f:[a,b]toBbb R$ that is differentiable on $(a,b)$. We are to prove MVT, i.e. to prove there exists a point $cin(a,b)$ such that $f'(c)=fracf(b)-f(a)b-a$, assuming that Rolle's theorem is true.
Define a new function $g:[a,b]toBbb R$ by $g(x)=f(x)-fracf(b)-f(a)b-ax$. Verify yourself that $g$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Verify also that $g(a)=g(b)$. So we can apply Rolle's theorem to this function $g$, to obtain a point $cin(a,b)$ such that $g'(c)=0$. Note that the derivative of $g$ is $g'(x)=f'(x)-fracf(b)-f(a)b-a$. The point $c$ also satisfies $f'(c)-fracf(b)-f(a)b-a=0$.
A proof of MVT used Rolle's theorem explicitly.
Consider a continuous function $f:[a,b]toBbb R$ that is differentiable on $(a,b)$. We are to prove MVT, i.e. to prove there exists a point $cin(a,b)$ such that $f'(c)=fracf(b)-f(a)b-a$, assuming that Rolle's theorem is true.
Define a new function $g:[a,b]toBbb R$ by $g(x)=f(x)-fracf(b)-f(a)b-ax$. Verify yourself that $g$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Verify also that $g(a)=g(b)$. So we can apply Rolle's theorem to this function $g$, to obtain a point $cin(a,b)$ such that $g'(c)=0$. Note that the derivative of $g$ is $g'(x)=f'(x)-fracf(b)-f(a)b-a$. The point $c$ also satisfies $f'(c)-fracf(b)-f(a)b-a=0$.
answered Oct 1 at 4:21
edm
2,8961424
2,8961424
add a comment |Â
add a comment |Â
up vote
4
down vote
You don't need any linear transformations in $mathbbR^2$; you can do this by subtracting a linear polynomial from your function (which is, itself, a shear map). Simply take your continuous function $f : [a, b] to mathbbR$ that is differentiable on $(a, b)$, and let $g(x) = f(x) - fracf(b) - f(a)b - a (x - a) - f(a)$. In particular $f(a) = f(b) = 0$, and when you apply Rolle's theorem to find $c in (a, b)$ such that $g'(c) = 0$, you'll find that
$$f'(c) = fracf(b)- f(a)b - a.$$
add a comment |Â
up vote
4
down vote
You don't need any linear transformations in $mathbbR^2$; you can do this by subtracting a linear polynomial from your function (which is, itself, a shear map). Simply take your continuous function $f : [a, b] to mathbbR$ that is differentiable on $(a, b)$, and let $g(x) = f(x) - fracf(b) - f(a)b - a (x - a) - f(a)$. In particular $f(a) = f(b) = 0$, and when you apply Rolle's theorem to find $c in (a, b)$ such that $g'(c) = 0$, you'll find that
$$f'(c) = fracf(b)- f(a)b - a.$$
add a comment |Â
up vote
4
down vote
up vote
4
down vote
You don't need any linear transformations in $mathbbR^2$; you can do this by subtracting a linear polynomial from your function (which is, itself, a shear map). Simply take your continuous function $f : [a, b] to mathbbR$ that is differentiable on $(a, b)$, and let $g(x) = f(x) - fracf(b) - f(a)b - a (x - a) - f(a)$. In particular $f(a) = f(b) = 0$, and when you apply Rolle's theorem to find $c in (a, b)$ such that $g'(c) = 0$, you'll find that
$$f'(c) = fracf(b)- f(a)b - a.$$
You don't need any linear transformations in $mathbbR^2$; you can do this by subtracting a linear polynomial from your function (which is, itself, a shear map). Simply take your continuous function $f : [a, b] to mathbbR$ that is differentiable on $(a, b)$, and let $g(x) = f(x) - fracf(b) - f(a)b - a (x - a) - f(a)$. In particular $f(a) = f(b) = 0$, and when you apply Rolle's theorem to find $c in (a, b)$ such that $g'(c) = 0$, you'll find that
$$f'(c) = fracf(b)- f(a)b - a.$$
answered Oct 1 at 4:20
Theo Bendit
14.7k12045
14.7k12045
add a comment |Â
add a comment |Â
up vote
1
down vote
Since none of the existing answers have explicitly pointed out your error:
For a function f, if it is differentiable/continuous over [a,b]/(a,b) etc. then there is a straight line between (a,f(a)) and (b,f(b)). You can rotate f around the origin until this straight line is parallel to the x-axis. [...]
No you cannot do this! You can apply Rolle's theorem to any differentiable function, but rotating the graph of a function in the cartesian plane may not result in a graph of another function, much less a differentiable function. This is the reason edm mentioned a shear in the comments, because a shear with an invariant vertical line is indeed the kind of transformation that would preserve differentiable functions.
add a comment |Â
up vote
1
down vote
Since none of the existing answers have explicitly pointed out your error:
For a function f, if it is differentiable/continuous over [a,b]/(a,b) etc. then there is a straight line between (a,f(a)) and (b,f(b)). You can rotate f around the origin until this straight line is parallel to the x-axis. [...]
No you cannot do this! You can apply Rolle's theorem to any differentiable function, but rotating the graph of a function in the cartesian plane may not result in a graph of another function, much less a differentiable function. This is the reason edm mentioned a shear in the comments, because a shear with an invariant vertical line is indeed the kind of transformation that would preserve differentiable functions.
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Since none of the existing answers have explicitly pointed out your error:
For a function f, if it is differentiable/continuous over [a,b]/(a,b) etc. then there is a straight line between (a,f(a)) and (b,f(b)). You can rotate f around the origin until this straight line is parallel to the x-axis. [...]
No you cannot do this! You can apply Rolle's theorem to any differentiable function, but rotating the graph of a function in the cartesian plane may not result in a graph of another function, much less a differentiable function. This is the reason edm mentioned a shear in the comments, because a shear with an invariant vertical line is indeed the kind of transformation that would preserve differentiable functions.
Since none of the existing answers have explicitly pointed out your error:
For a function f, if it is differentiable/continuous over [a,b]/(a,b) etc. then there is a straight line between (a,f(a)) and (b,f(b)). You can rotate f around the origin until this straight line is parallel to the x-axis. [...]
No you cannot do this! You can apply Rolle's theorem to any differentiable function, but rotating the graph of a function in the cartesian plane may not result in a graph of another function, much less a differentiable function. This is the reason edm mentioned a shear in the comments, because a shear with an invariant vertical line is indeed the kind of transformation that would preserve differentiable functions.
answered Oct 1 at 12:25
user21820
36.8k441143
36.8k441143
add a comment |Â
add a comment |Â
up vote
-1
down vote
Not really.
Rolle's theorum is a special case of Legrange's MVT. Not the other way around. So just by assuming Rolle's theorum, MVT is not automatically proved. We do of course use RT in the MVT proof. But just because Rolle's theorum is true, MVT need'nt be.
If you are wondering why we use this special case to prove the MVT, it is because Rolle's theorum appeared first.
New contributor
add a comment |Â
up vote
-1
down vote
Not really.
Rolle's theorum is a special case of Legrange's MVT. Not the other way around. So just by assuming Rolle's theorum, MVT is not automatically proved. We do of course use RT in the MVT proof. But just because Rolle's theorum is true, MVT need'nt be.
If you are wondering why we use this special case to prove the MVT, it is because Rolle's theorum appeared first.
New contributor
add a comment |Â
up vote
-1
down vote
up vote
-1
down vote
Not really.
Rolle's theorum is a special case of Legrange's MVT. Not the other way around. So just by assuming Rolle's theorum, MVT is not automatically proved. We do of course use RT in the MVT proof. But just because Rolle's theorum is true, MVT need'nt be.
If you are wondering why we use this special case to prove the MVT, it is because Rolle's theorum appeared first.
New contributor
Not really.
Rolle's theorum is a special case of Legrange's MVT. Not the other way around. So just by assuming Rolle's theorum, MVT is not automatically proved. We do of course use RT in the MVT proof. But just because Rolle's theorum is true, MVT need'nt be.
If you are wondering why we use this special case to prove the MVT, it is because Rolle's theorum appeared first.
New contributor
New contributor
answered Oct 1 at 12:25
ja.varkey
1
1
New contributor
New contributor
add a comment |Â
add a comment |Â
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2
If the question is "Can the mean value theorem be proved using Rolle's theorem?" the answer is "Yes." However, I don't see what this has to do with linear algebra, and I am very confused by your exposition.
â Xander Henderson
Oct 1 at 4:06
1
Did you read any notes or books on a proof of MVT? It should be clear that Rolle's theorem is used in any proof. If it is not clear to you, I can write an outline for you.
â edm
Oct 1 at 4:07
3
This is indeed how the MVT is often proved.
â Bungo
Oct 1 at 4:07
Perhaps what you mean by "rotate and stretch" is shear mapping.
â edm
Oct 1 at 4:15
2
@Matt There's a problem with your idea of rotating: it may cause the rotated graph to no longer be a function!
â Theo Bendit
Oct 1 at 4:40