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mathmath333

  • one year ago

Find the minimum value of

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  1. mathmath333
    • one year ago
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    \(\large \color{black}{\begin{align} |x-1|+|x-2|+|x-3|+\cdots \ \cdots+|x-75|,\ \ x\in\mathbb{R}\hspace{.33em}\\~\\ \end{align}}\)

  2. anonymous
    • one year ago
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    thats a hard one

  3. anonymous
    • one year ago
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    maybe -77 ?

  4. mathmath333
    • one year ago
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    absolute value is always +

  5. anonymous
    • one year ago
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    yes true true

  6. perl
    • one year ago
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    I get around 1406

  7. mathmath333
    • one year ago
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    how

  8. mathmath333
    • one year ago
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    thats right btw

  9. perl
    • one year ago
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    that gives you x = 38

  10. perl
    • one year ago
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    thats correct

  11. ParthKohli
    • one year ago
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    Yeah, I meant 38.

  12. ParthKohli
    • one year ago
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    \[37\cdot 38 = 1406\]

  13. perl
    • one year ago
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    mathmath, I used computer software and did it by brute force. not elegantly

  14. ganeshie8
    • one year ago
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    38 is the median value of integers 1-75

  15. ganeshie8
    • one year ago
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    it works because of symmetry but we need to prove it i guess

  16. mathmath333
    • one year ago
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    u did (1+75)/2

  17. ParthKohli
    • one year ago
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    \[|x - 1| + \cdots + |x - 75|\ge |75x - 75\cdot 38| = 75|x - 38| \]Equality occurs at \(x=38\).

  18. ParthKohli
    • one year ago
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    Again, wrong method to do it. =_=

  19. mathmath333
    • one year ago
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    what is wrong

  20. ParthKohli
    • one year ago
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    The answer is correct, but it's the wrong approach.

  21. mathmath333
    • one year ago
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    in case it was this then how we do it \(\large \color{black}{\begin{align} |x+0|+|x-1|+|x-2|+|x-3|+\cdots \quad \cdots+|x-75|,\ \ x\in\mathbb{R}\hspace{.33em}\\~\\ \end{align}}\)

  22. ParthKohli
    • one year ago
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    This function is continuous and also symmetric about \(38\), so that is where the minimum should occur as there is no maximum.

  23. perl
    • one year ago
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    if you take the derivative of that function it is negative for x < 38 and positive for x > 38

  24. ParthKohli
    • one year ago
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    Derivatives with absolute values?

  25. mathmath333
    • one year ago
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    i dont know much calculus

  26. perl
    • one year ago
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    $$\Large |x| =\sqrt{x^2}$$

  27. perl
    • one year ago
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    This is not the optimal approach, maybe a last resort.

  28. ParthKohli
    • one year ago
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    Yeah, I think the best way to explain it is \(f(38+k) = f(38 - k)\)

  29. mathmath333
    • one year ago
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    i was asking for this edited question do i here also substitute \(38\) \(\large \color{black}{\begin{align} |x-0|+|x-1|+|x-2|+|x-3|+\cdots \quad \cdots+|x-75|,\ \ x\in\mathbb{R}\hspace{.33em}\\~\\ \end{align}}\)

  30. imqwerty
    • one year ago
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    we take some cases 1) suppose x=<0 then all the mods will open as negative and we'll get a big number 2) suppose x>=75 then all mods will open positive and in this case we'll get a big number 3)if 0<x<75 then some mods will open positive and some negative nd thus the negative ones and positive ones will cancel each other nd we'll get a smaller value so we take a middle value from the numbers 1-75 i.e (1+75)/2 = 38 then then'll u must be able to solve it

  31. imqwerty
    • one year ago
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    yes @mathmath333 u have to put x = 38 in that equation

  32. ParthKohli
    • one year ago
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    Observe that \(f(37.5 + k) = f(37.5 - k) \) so min should occur at \(37.5\)

  33. perl
    • one year ago
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    f(37.5) = 1406.5 f(38) = 1406

  34. ParthKohli
    • one year ago
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    ooo.

  35. imqwerty
    • one year ago
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    plugging x = 38 u get the value of equation as 1406

  36. ganeshie8
    • one year ago
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    Nice! just nitpicking on your latest reply OK ;p below is symmetric but the min value occurs somewhere else |dw:1434029538657:dw|

  37. mathmath333
    • one year ago
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    but calculated 1406 for previous question adding \(|38-0+(1406) | >1406\)

  38. ganeshie8
    • one year ago
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    *PK

  39. ParthKohli
    • one year ago
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    Yeah, I saw that \(f(37) = f(38)\) and somehow came to the conclusion that it should be minimum at \(37.5\) Sorry abou thtat

  40. ParthKohli
    • one year ago
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    My intuition somehow always manages to stay one step behind me.

  41. ParthKohli
    • one year ago
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    \[f(37) = f(38)\] Scared me for a moment there. I was about to go to sleep.

  42. ganeshie8
    • one year ago
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    nope, looks ur intuition works perfectly in present problem!

  43. ParthKohli
    • one year ago
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    Nah, I got the \(|x| + |x- 1| + \cdots + |x- 74 | + |x- 75|\) problem wrong exactly because of the reason you posted.

  44. ParthKohli
    • one year ago
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    I have a neat problem about absolute values. Hold on. Gonna post it.

  45. Pawanyadav
    • one year ago
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    Mathematically can't solve this problem

  46. perl
    • one year ago
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    can you generalize the problem find the min of $$ \Large \sum_{k=0}^n | x - (a+k)| $$

  47. ParthKohli
    • one year ago
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    It depends on the number of terms being odd/even. We covered both cases. In both cases, we see the median.

  48. mathmath333
    • one year ago
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    how can i solve this with wolfram

  49. perl
    • one year ago
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    it exceeds wolframs computational abilities but the code is https://www.wolframalpha.com/input/?i=Min%5B+Sum%5BAbs%5Bx+-+k%5D%2C+%7Bk%2C+1%2C+75%7D%5D%5D

  50. mathmath333
    • one year ago
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    also this is not working https://www.desmos.com/calculator/bpayybqovh

  51. perl
    • one year ago
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    using the graph you can estimate that the minimum is between 35 and 40. then plug this number into wolfram manually https://www.wolframalpha.com/input/?i=+Sum%5BAbs%5B37-+k%5D%2C+%7Bk%2C+1%2C+75%7D%5D%5D%2C++Sum%5BAbs%5B38-+k%5D%2C+%7Bk%2C+1%2C+75%7D%5D%5D%2C+Sum%5BAbs%5B39-+k%5D%2C+%7Bk%2C+1%2C+75%7D%5D%5D

  52. mathmath333
    • one year ago
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    wait desmos graph gave \( \Large y_{min}=1444\)

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  53. perl
    • one year ago
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    Hmm

  54. perl
    • one year ago
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    Oh you have to zoom out

  55. mathmath333
    • one year ago
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    so \(1406+38=1444\) haha indeed for the second problem \(y=38\) was correct @imqwerty u were correct indeed

  56. perl
    • one year ago
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    just change the code a little bit http://prntscr.com/7fqk52

  57. mathmath333
    • one year ago
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    so parthkohli answer of 37.5 was incorrct ?

  58. perl
    • one year ago
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    the correct answer is 38

  59. perl
    • one year ago
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    https://www.desmos.com/calculator/iehwd6yzbg

  60. mathmath333
    • one year ago
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    i m still confused why 38 works on both cases

  61. perl
    • one year ago
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    you're talking about the other problem

  62. perl
    • one year ago
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    i caught a glimpse of it,

  63. mathmath333
    • one year ago
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    for both problem

  64. perl
    • one year ago
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    g(x)= sum | x - k | , k=0..75 f(x) = sum | x - k | , k=1..75 agreed?

  65. mathmath333
    • one year ago
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    yes

  66. perl
    • one year ago
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    g(37) = g(38) = 1444 f(38) = 1406

  67. perl
    • one year ago
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    According to my calculations g(37)=g(37.5) = g(38) there is a flat line there http://prntscr.com/7fqoum

  68. perl
    • one year ago
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    Therefore it is not wrong to say g(37.5) is the minimum. It is. So is g(37) and g(38)

  69. perl
    • one year ago
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    In the case f(x), we have corner , not a flat line, at the minimum.

  70. perl
    • one year ago
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    If you zoom in enough

  71. mathmath333
    • one year ago
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    how can u prove \(g(37.5)=g(38)\)

  72. mathmath333
    • one year ago
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    ok i saw that by zooming .

  73. perl
    • one year ago
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    and the other case is this http://prntscr.com/7fqqtn

  74. mathmath333
    • one year ago
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    graph shows \(g(37\leq x\leq 38)=same\)

  75. perl
    • one year ago
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    I hope that's not too confusing :)

  76. perl
    • one year ago
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    Example: {1,2,3,4,5} <--- median is 3 {1,2,3,4} <--- median is 2.5 , the average of 2 and 3

  77. mathmath333
    • one year ago
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    actually when the numbers are even like for 1,2,3,4, then median is \(2\leq x\leq 3\) any value of inequality of x will work

  78. perl
    • one year ago
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    yes

  79. perl
    • one year ago
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    I am curious did you mean to put a plus sign here or was this a different problem. http://prntscr.com/7fr0o7

  80. mathmath333
    • one year ago
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    lol that was typo it should be this \(\large \color{black}{\begin{align} \left(\sum_{k=0}^{75}|x-k| \right )\hspace{.33em}\\~\\ \end{align}}\)

  81. IrishBoy123
    • one year ago
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    FWIW, in Python x sum of abs's 36.0 1410.0 36.5 1408.5 37.0 1407.0 37.5 1406.5 38.0 1406.0 38.5 1406.5 39.0 1407.0 39.5 1408.5 40.0 1410.0

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