Question

Find the minimum value of

Answer 1

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

Answer 2

thats a hard one

Answer 3

maybe -77 ?

Answer 4

absolute value is always +

Answer 5

yes true true

Answer 6

I get around 1406

Answer 7

how

Answer 8

thats right btw

Answer 9

that gives you x = 38

Answer 10

thats correct

Answer 11

Yeah, I meant 38.

Answer 12

\[37\cdot 38 = 1406\]

Answer 13

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

Answer 14

38 is the median value of integers 1-75

Answer 15

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

Answer 16

u did (1+75)/2

Answer 17

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

Answer 18

Again, wrong method to do it. =_=

Answer 19

what is wrong

Answer 20

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

Answer 21

in case it was this
then how we do it

\(\large \color{black}{\begin{align} |x+0|+|x-1|+|x-2|+|x-3|+\cdots \quad \cdots+|x-75|,\ \ x\in\mathbb{R}\hspace{.33em}\\~\\
\end{align}}\)

Answer 22

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

Answer 23

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

Answer 24

Derivatives with absolute values?

Answer 25

i dont know much calculus

Answer 26

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

Answer 27

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

Answer 28

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

Answer 29

i was asking for this edited question do i here also substitute \(38\)

\(\large \color{black}{\begin{align} |x-0|+|x-1|+|x-2|+|x-3|+\cdots \quad \cdots+|x-75|,\ \ x\in\mathbb{R}\hspace{.33em}\\~\\
\end{align}}\)

Answer 30

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

Answer 31

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

Answer 32

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

Answer 33

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

Answer 34

ooo.

Answer 35

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

Answer 36

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