mathmath333
  • mathmath333
Find the minimum value of
Mathematics
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
mathmath333
  • mathmath333
\(\large \color{black}{\begin{align} |x-1|+|x-2|+|x-3|+\cdots \ \cdots+|x-75|,\ \ x\in\mathbb{R}\hspace{.33em}\\~\\ \end{align}}\)
anonymous
  • anonymous
thats a hard one
anonymous
  • anonymous
maybe -77 ?

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

mathmath333
  • mathmath333
absolute value is always +
anonymous
  • anonymous
yes true true
perl
  • perl
I get around 1406
mathmath333
  • mathmath333
how
mathmath333
  • mathmath333
thats right btw
perl
  • perl
that gives you x = 38
perl
  • perl
thats correct
ParthKohli
  • ParthKohli
Yeah, I meant 38.
ParthKohli
  • ParthKohli
\[37\cdot 38 = 1406\]
perl
  • perl
mathmath, I used computer software and did it by brute force. not elegantly
ganeshie8
  • ganeshie8
38 is the median value of integers 1-75
ganeshie8
  • ganeshie8
it works because of symmetry but we need to prove it i guess
mathmath333
  • mathmath333
u did (1+75)/2
ParthKohli
  • ParthKohli
\[|x - 1| + \cdots + |x - 75|\ge |75x - 75\cdot 38| = 75|x - 38| \]Equality occurs at \(x=38\).
ParthKohli
  • ParthKohli
Again, wrong method to do it. =_=
mathmath333
  • mathmath333
what is wrong
ParthKohli
  • ParthKohli
The answer is correct, but it's the wrong approach.
mathmath333
  • mathmath333
in case it was this then how we do it \(\large \color{black}{\begin{align} |x+0|+|x-1|+|x-2|+|x-3|+\cdots \quad \cdots+|x-75|,\ \ x\in\mathbb{R}\hspace{.33em}\\~\\ \end{align}}\)
ParthKohli
  • ParthKohli
This function is continuous and also symmetric about \(38\), so that is where the minimum should occur as there is no maximum.
perl
  • perl
if you take the derivative of that function it is negative for x < 38 and positive for x > 38
ParthKohli
  • ParthKohli
Derivatives with absolute values?
mathmath333
  • mathmath333
i dont know much calculus
perl
  • perl
$$\Large |x| =\sqrt{x^2}$$
perl
  • perl
This is not the optimal approach, maybe a last resort.
ParthKohli
  • ParthKohli
Yeah, I think the best way to explain it is \(f(38+k) = f(38 - k)\)
mathmath333
  • mathmath333
i was asking for this edited question do i here also substitute \(38\) \(\large \color{black}{\begin{align} |x-0|+|x-1|+|x-2|+|x-3|+\cdots \quad \cdots+|x-75|,\ \ x\in\mathbb{R}\hspace{.33em}\\~\\ \end{align}}\)
imqwerty
  • imqwerty
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
imqwerty
  • imqwerty
yes @mathmath333 u have to put x = 38 in that equation
ParthKohli
  • ParthKohli
Observe that \(f(37.5 + k) = f(37.5 - k) \) so min should occur at \(37.5\)
perl
  • perl
f(37.5) = 1406.5 f(38) = 1406
ParthKohli
  • ParthKohli
ooo.
imqwerty
  • imqwerty
plugging x = 38 u get the value of equation as 1406
ganeshie8
  • ganeshie8
Nice! just nitpicking on your latest reply OK ;p below is symmetric but the min value occurs somewhere else |dw:1434029538657:dw|
mathmath333
  • mathmath333
but calculated 1406 for previous question adding \(|38-0+(1406) | >1406\)
ganeshie8
  • ganeshie8
*PK
ParthKohli
  • ParthKohli
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
ParthKohli
  • ParthKohli
My intuition somehow always manages to stay one step behind me.
ParthKohli
  • ParthKohli
\[f(37) = f(38)\] Scared me for a moment there. I was about to go to sleep.
ganeshie8
  • ganeshie8
nope, looks ur intuition works perfectly in present problem!
ParthKohli
  • ParthKohli
Nah, I got the \(|x| + |x- 1| + \cdots + |x- 74 | + |x- 75|\) problem wrong exactly because of the reason you posted.
ParthKohli
  • ParthKohli
I have a neat problem about absolute values. Hold on. Gonna post it.
Pawanyadav
  • Pawanyadav
Mathematically can't solve this problem
perl
  • perl
can you generalize the problem find the min of $$ \Large \sum_{k=0}^n | x - (a+k)| $$
ParthKohli
  • ParthKohli
It depends on the number of terms being odd/even. We covered both cases. In both cases, we see the median.
mathmath333
  • mathmath333
how can i solve this with wolfram
perl
  • perl
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
mathmath333
  • mathmath333
also this is not working https://www.desmos.com/calculator/bpayybqovh
perl
  • perl
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
mathmath333
  • mathmath333
wait desmos graph gave \( \Large y_{min}=1444\)
1 Attachment
perl
  • perl
Hmm
perl
  • perl
Oh you have to zoom out
mathmath333
  • mathmath333
so \(1406+38=1444\) haha indeed for the second problem \(y=38\) was correct @imqwerty u were correct indeed
perl
  • perl
just change the code a little bit http://prntscr.com/7fqk52
mathmath333
  • mathmath333
so parthkohli answer of 37.5 was incorrct ?
perl
  • perl
the correct answer is 38
perl
  • perl
https://www.desmos.com/calculator/iehwd6yzbg
mathmath333
  • mathmath333
i m still confused why 38 works on both cases
perl
  • perl
you're talking about the other problem
perl
  • perl
i caught a glimpse of it,
mathmath333
  • mathmath333
for both problem
perl
  • perl
g(x)= sum | x - k | , k=0..75 f(x) = sum | x - k | , k=1..75 agreed?
mathmath333
  • mathmath333
yes
perl
  • perl
g(37) = g(38) = 1444 f(38) = 1406
perl
  • perl
According to my calculations g(37)=g(37.5) = g(38) there is a flat line there http://prntscr.com/7fqoum
perl
  • perl
Therefore it is not wrong to say g(37.5) is the minimum. It is. So is g(37) and g(38)
perl
  • perl
In the case f(x), we have corner , not a flat line, at the minimum.
perl
  • perl
If you zoom in enough
mathmath333
  • mathmath333
how can u prove \(g(37.5)=g(38)\)
mathmath333
  • mathmath333
ok i saw that by zooming .
perl
  • perl
and the other case is this http://prntscr.com/7fqqtn
mathmath333
  • mathmath333
graph shows \(g(37\leq x\leq 38)=same\)
perl
  • perl
I hope that's not too confusing :)
perl
  • perl
Example: {1,2,3,4,5} <--- median is 3 {1,2,3,4} <--- median is 2.5 , the average of 2 and 3
mathmath333
  • mathmath333
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
perl
  • perl
yes
perl
  • perl
I am curious did you mean to put a plus sign here or was this a different problem. http://prntscr.com/7fr0o7
mathmath333
  • mathmath333
lol that was typo it should be this \(\large \color{black}{\begin{align} \left(\sum_{k=0}^{75}|x-k| \right )\hspace{.33em}\\~\\ \end{align}}\)
IrishBoy123
  • IrishBoy123
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
1 Attachment

Looking for something else?

Not the answer you are looking for? Search for more explanations.