`f(x)=|a x-b|+c|x|AAx in (-oo,oo),` where `a >0, b >0,c > 0.` Find the condition if `f(x)` attains the minimum value only at one point.

Explanation

We have `f(x) =|ax-b|+c|x|forall`x in (-00,00),Where `agt0, bgt0`,0.Find the condition if f(x) attains minimum value only at one point.
`{b-(a+c)x,xlt0`
`b+(c-a)x,0lexlt(b)/(a)`
`(a+c)x+b, xge(b)/(a)`
slope of y=b -(a+c)x is negative
Slope of y=(a+c)x+b positive
slope of y =+(c-a)x is negative zero or positive if `clta,c=a` and `clt` a repsectively.
We have following possible graphs of given function

In fig(i) function has one point of minima at x=0
In fig(ii) functions has infinite points of minima at `x=c//a`
Hence function agains minimum value at only one point of `cnea`.