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if a,b,c are distinct positive numbers , then the expression (b+c-a)(c+a-b)(a+b-c)-abc is positive or negative........?

6 years ago


Answers : (1)


Dear nishant

let x ,y,z are three positive number

so use AM ≥GM

 x+y ≥2(xy)1/2

 y+z ≥2(yz)1/2

 z+x ≥2(zx)1/2

 so (x+y)(y+z)(z+x)≥2(xy)1/22(yz)1/22(zx)1/2

  or (x+y)(y+z)(z+x)≥8(xyz)  

 now put x=b+c-a 

           y= c+a-b

           z= a+b-c

  or x+y =2c  , y+z =2a , z+x =2b

(2c)(2a)(2b)    ≥8(b+c-a )( c+a-b)( a+b-c)

  abc      ≥(b+c-a )( c+a-b)( a+b-c)

 (b+c-a )( c+a-b)( a+b-c) -abc ≤ 0

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6 years ago

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