Question

minimum of a function

In the following roblem,
find the least value of the function:
f(x)= +|x-a| +|x-b| + |x-c|
where x is real and, a<B<C< real fixed are a,b,c,d>
i tried using the AM-GM inequality but it was of no use. how should i start?

Answer 1

Notice that f(x) is the sum of three non-negative functions
|x-a|,
|x-b|, |x-c|
having minimums at a,b,c respectively.

Also notice that
since a<b<c

f(a) = |a-a| + |a-b| + |a-c| = b-a+c-a =
b+c-2a

f(b) = |b-a| + |b-b| + |b-c| = b-a+c-b = c-a

f(c) = |c-a| +
|c-b| + |c-c| = c-a+c-b = 2c-a-b

Since
f(a) = b-a + c-a = b-a +
f(b)
f(c) = c-a + c-b = f(b) + c-b

Thus f(b) < f(a), and f(b)
< f(c).

We claim that x=b is the minimum of f, so
min f = f(b) =
c-a.

It remains to show that for any number x distinct from a,b,c we have
that
f(x) > f(b) = c-a.

Consider the following 4
cases:

1) If x<a<b<c, then

|x-c| > c-a, whence


f(x)= |x-a| +|x-b| + |x-c| > c-a = f(b)


2) If
a<x<b<c, then

|x-a| + |x-c| = c-a,
and

|x-b|>0,
whence

f(x)= |x-a| +|x-b| + |x-c| > c-a = f(b)



3) If a<b<x<c, then

|x-a| + |x-c| = c-a,
and

|x-b|>0,
whence

f(x)= |x-a| +|x-b| + |x-c| > c-a = f(b)



4) If a<b<c<x, then

|x-a| > c-a,
whence

f(x)= |x-a| +|x-b| + |x-c| > c-a = f(b)


Thus
min(f) =
f(b) = c-a.