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73,044 Views • Jun 6, 2024 • Click to toggle off description
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Thanks!
It turns out that this theorem is equivalent to the Arithmetic Mean-Geometric Mean inequality. The equivalence is implied by the following two proofs: • Visual Proof of AM-GM Inequality I
• x + 1/x is greater than or equal to 2...
And here are two alternate ways to prove this fact via the Pythagorean Theorem and triangle areas:
• x plus 1/x is greater than or equal t...
• x plus 1/x is greater than or equal t...
This animation is based on a proof by Roger B. Nelsen from the December 1994 issue of Mathematics Magazine, page 374 (doi.org/10.2307/2690999). You can find this here too: www.maa.org/sites/default/files/Nelsen99959975.pdf
#math #manim #visualproof #mathvideo #geometry #mathshorts #algebra #mtbos #animation #theorem #pww #proofwithoutwords #proof #iteachmath #inequality #calculus #derivative #tangentline #slope #amgm
To learn more about animating with manim, check out:
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Views : 73,044
Genre: Education
License: Standard YouTube License
Uploaded At Jun 6, 2024 ^^
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RYD date created : 2024-09-13T07:51:26.114224Z
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41 Comments
Top Comments of this video!! :3
That's a creative use of calculus ngl
83 |
another method using calculus:
compaire 1/x + x and 2 so we do
1/x + x - 2
= (x² - 2x + 1)/x
= (x - 1)²/x
(x-1)² always positive
and we have x > 0
That means (x-1)²/x ≥ 0
1/x + x - 2 ≥ 0
1/x + x ≥ 2
but I liked your method
29 |
The function f(x) = x + 1/x is an example of a function that is invariant under inversion, meaning it is unchanged when x is replaced with 1/x.
Another way of saying the same thing is that f(x) = x + 1/x is a solution to the functional equation f(x) = f(1/x). A function is invariant under inversion if and only if it is a solution to this functional equation.
It is a kind of symmetry similar to functions that satisfy f(-x)=f(x), called "even" functions, which are symmetrical about the y-axis, except here the function on the interval (0, 1] is symmetric with the function on [1, ∞).
There is a relationship between even functions and functions that are invariant under inversion. Hint: think about how log functions transform multiplication into addition.
61 |
I wished you'd give us a little more time at the end of the video to contemplate your question, but it helped to watch the video a second time.
15 |
There's a nice visual proof for the AM-GM inequality, using the curveness of ln (or e^x, doesn't really matter) like here.
11 |
wow, that's a way to prove it!
so cool!
6 |
I was just learning about this function, YouTube algorithm works in mysterious ways
2 |
Such a beautiful proof.
1 |
Another way to prove this :
(x-1)^2 >= 0 (because a square is always positive)
x^2-2x+1 >= 0
x^2+1 >= 2x
Now divide by x, the inequality won’t change because x>0
x+1/x >= 2
The equality happens when x=1
|
Fun fact: the area bounded by the tangent line of f(x) = 1/x at any point x = a with the positive x and y axes is always 2.
|
Considering its given that x>0 couldnt you multiply both sides by x without any consequence?
2 |
Just use AM GM inequality......
1 |
Beautiful
1 |
Could also be proved using the property AM is greater than or equal to GM
|
I have never seen this, and now I can’t not see it!😊
2 |
Hi! I’m curious, What software do you use to edit/create these videos?
1 |
COOL
1 |
yay jensen's
|
|N+1/n|≥2 for n in R
|
@yurfwendforju
5 months ago
bro that's crazy. I never looked at functions like this
187 |