Find the minimum value of

- mathmath333

Find the minimum value of

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- 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

thats a hard one

- anonymous

maybe -77 ?

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## More answers

- mathmath333

absolute value is always +

- anonymous

yes true true

- perl

I get around 1406

- mathmath333

how

- mathmath333

thats right btw

- perl

that gives you x = 38

- perl

thats correct

- ParthKohli

Yeah, I meant 38.

- ParthKohli

\[37\cdot 38 = 1406\]

- perl

mathmath, I used computer software and did it by brute force. not elegantly

- ganeshie8

38 is the median value of integers 1-75

- ganeshie8

it works because of symmetry but we need to prove it i guess

- mathmath333

u did (1+75)/2

- ParthKohli

\[|x - 1| + \cdots + |x - 75|\ge |75x - 75\cdot 38| = 75|x - 38| \]Equality occurs at \(x=38\).

- ParthKohli

Again, wrong method to do it. =_=

- mathmath333

what is wrong

- ParthKohli

The answer is correct, but it's the wrong approach.

- 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

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

- perl

if you take the derivative of that function it is negative for x < 38 and positive for x > 38

- ParthKohli

Derivatives with absolute values?

- mathmath333

i dont know much calculus

- perl

$$\Large |x| =\sqrt{x^2}$$

- perl

This is not the optimal approach, maybe a last resort.

- ParthKohli

Yeah, I think the best way to explain it is \(f(38+k) = f(38 - k)\)

- 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

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

yes @mathmath333 u have to put x = 38 in that equation

- ParthKohli

Observe that \(f(37.5 + k) = f(37.5 - k) \) so min should occur at \(37.5\)

- perl

f(37.5) = 1406.5
f(38) = 1406

- ParthKohli

ooo.

- imqwerty

plugging x = 38 u get the value of equation as 1406

- ganeshie8

Nice! just nitpicking on your latest reply OK ;p
below is symmetric but the min value occurs somewhere else
|dw:1434029538657:dw|

- mathmath333

but calculated 1406 for previous question adding \(|38-0+(1406) | >1406\)

- ganeshie8

*PK

- 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

My intuition somehow always manages to stay one step behind me.

- ParthKohli

\[f(37) = f(38)\]
Scared me for a moment there. I was about to go to sleep.

- ganeshie8

nope, looks ur intuition works perfectly in present problem!

- ParthKohli

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

- ParthKohli

I have a neat problem about absolute values. Hold on. Gonna post it.

- Pawanyadav

Mathematically can't solve this problem

- perl

can you generalize the problem
find the min of $$ \Large \sum_{k=0}^n | x - (a+k)| $$

- ParthKohli

It depends on the number of terms being odd/even. We covered both cases. In both cases, we see the median.

- mathmath333

how can i solve this with wolfram

- 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

also this is not working
https://www.desmos.com/calculator/bpayybqovh

- 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

wait desmos graph gave
\( \Large y_{min}=1444\)

##### 1 Attachment

- perl

Hmm

- perl

Oh you have to zoom out

- mathmath333

so \(1406+38=1444\) haha indeed for the second problem \(y=38\) was correct
@imqwerty u were correct indeed

- perl

just change the code a little bit
http://prntscr.com/7fqk52

- mathmath333

so parthkohli answer of 37.5 was incorrct ?

- perl

the correct answer is 38

- perl

https://www.desmos.com/calculator/iehwd6yzbg

- mathmath333

i m still confused why 38 works on both cases

- perl

you're talking about the other problem

- perl

i caught a glimpse of it,

- mathmath333

for both problem

- perl

g(x)= sum | x - k | , k=0..75
f(x) = sum | x - k | , k=1..75
agreed?

- mathmath333

yes

- perl

g(37) = g(38) = 1444
f(38) = 1406

- perl

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

- perl

Therefore it is not wrong to say g(37.5) is the minimum. It is. So is g(37) and g(38)

- perl

In the case f(x), we have corner , not a flat line, at the minimum.

- perl

If you zoom in enough

- mathmath333

how can u prove \(g(37.5)=g(38)\)

- mathmath333

ok i saw that by zooming .

- perl

and the other case is this
http://prntscr.com/7fqqtn

- mathmath333

graph shows
\(g(37\leq x\leq 38)=same\)

- perl

I hope that's not too confusing :)

- 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

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

yes

- perl

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

- 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

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

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