\(\frac{|a+b|}{|a|+|b|} + \frac{|b+c|}{|b|+|c|} +\frac{|c+a|}{|c|+|a|}= M\)
What is the difference between maximum and minimum value of M?

Minimum value of the expression is 1 when a=-b ; b=-c; a=-b=c

Maximum value of expression is when
a,b & c have same sign
The maximum value of the expression =3

Difference = 3-1=2

IMO C

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Highlighted part is not correct. It's true only when a=b=c=0 (this case is not possible in this question)
if a=-b and b= -c
then a must be equal to c

At a=-b=c, M has the minimum value that is 1.

Hmmmmm, why can't the expresion value be equal to 0?

Corrected. Thanks


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No restrictions are given for values of a, b or c. so to simplify this, lets assume the value to be |a| =|b|=|c| = 1

we can see that, |a| + |b| = |b| + |c| = |c| + |a| = 2.... Hence all bases are same. The max. and min value totally depends on numerators now.

Maximum value can be obtained when sign of all numbers is same (we get all additions inside modulus and hence largest value). when all a, b and c all are positive or negative we will get 6 in the numerator. So the largest possible value of M would be 6/2 = 3

Smallest possible value can be obtained when one of a, b and c is having different sign while other two have same sign. (By this way there would be 2 subtractions inside modulus and 1 addition) By considering any one of a, b or c to be negative, we get 2 in the numerator. So the smallest number than can be obtained is 2/2 = 1

So the difference between largest and the smallest value is 3 -1 = 2, Hence IMO (C)

Hi Apo_orv

Please let me know for which values of a,b or c , the expression will be 0?????

Thanks

Can somebody please help me?

I am stuck how M can be 1 at minimum. given a = -b = c, the first two terms are euqal to 0, right?
-(a+b)/a+b = -a-b/a+b = -a+a/a+b = -a +a = 0 hence 0,

The 3th term is different :
-(c+a)/c+a => -c-a/c+a = -2a/2a = -1. So how do you get 1 as a result?

Well

yur resoning is incorrect for above highlighted part

i would rather suggest to try some real numbers ...

Hope it helps

Thank you very much for the reply, but im stuck. When i set in real numbres, how would i choose them?
We are saying that a = -b = c
So a =1, then b =-1 and c =1?

a =1, then b =-1 and c =1?

Yes ...u can take any...