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nishant kumar agarwal Grade:
        

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)

Badiuddin askIITians.ismu Expert
147 Points
										

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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Badiuddin

6 years ago
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