Question

can anyone suggest me with the graph of f(x)= |x| + |x-1| + |x-2| +|x-3|

Answer 1

for x >=3, the graph is a straight line : 4x - 6

for x <=0, the graph is a straight line : -4x + 6

need to study for x in between (0, 3)

Answer 2

yeah that what I need, I remember someone taught me that graph... and need its minimum value and that x too

Answer 3

used graphing tool on my mac :)

Answer 4

we are not allowed to use electronic devices in exam :P @Sepeario

Answer 5

Ok I would start with the fact that y must be at least 3.

Answer 6

@rational what do you say.. define it in every interval then plot its graph?

Answer 7

I think the minimum values for `|x| + x-1|+|x-2|+...` is achieved when x is the median of `0, 1, 2, ... `

Answer 8

for example, the median of `0, 1, 2, 3, 4` is 2
therefore the minimum value of `|x|+|x-1|+|x-2|+|x-3|+|x-4|` is achieved when `x=2`

Answer 9

but can this be generalized as for x E [1.2] the value is constant 4

Answer 10

another example, the median of `2, 5, 6, 10, 12` is 6
therefore the minimum value of `|x-2|+|x-5|+|x-6|+|x-10|+|x-12|` must be achieved at `x=6`

Answer 11

I think you will get a range of "x" for minimum when the number of terms is even

Answer 12

but if the numbers are not close? then for the minimum value the biggest one have to be zero

Answer 13

when the number of terms is odd, there will be a exact median, so u get only one "x" where the minimum is achieved

Answer 14

I think it doesn't matter,
the minimum is always achieved for the median value of x

Answer 15

can you provide an example of where you think it could fail ?

Answer 16

another example, the median of `2, 5, 6, 10, 10000000` is 6
therefore the minimum value of `|x-2|+|x-5|+|x-6|+|x-10|+|x-10000000|` must be achieved at `x=6`

Answer 17

yeah you are correct..

Answer 18

thanks :) @rational

Answer 19

np

Answer 20

.