How does one prove that (a/sqrt(a 2 +8bc)) + (b/sqrt(b 2 +8ac)) + (c/sqrt(c 2 +8ab)) = 1 ? sqrt() is the square root function

Question

How does one prove that (a/sqrt(a 2 +8bc)) + (b/sqrt(b 2 +8ac)) + (c/sqrt(c 2 +8ab)) >= 1 ? sqrt() is the square root function

Badiuddin askIITians.ismu Expert · Answer

Dear Mardava RJgpl ((a/sqrt(a 2 +8bc)) + (b/sqrt(b 2 +8ac)) + (c/sqrt(c 2 +8ab)) >= 1 USe AM >=GM ((a/sqrt(a 2 +8bc)) + (b/sqrt(b 2 +8ac)) + (c/sqrt(c 2 +8ab))/3 >=( (a/sqrt(a 2 +8bc)) (b/sqrt(b 2 +8ac)) (c/sqrt(c 2 +8ab)) 1/3 ............................1 Now use again AM>= GM Let three number a 2 ,4bc,4bc (a2 +4bc +4bc)/3 >=(16a 2 b 2 c 2 ) 1/3 (a2 +8bc ) >=3(16a 2 b 2 c 2 ) 1/3 or sqrt (a2 +8bc )>=&radic;3 (4abc) 1/3 use this result in equation 1 ((a/sqrt(a 2 +8bc)) + (b/sqrt(b 2 +8ac)) + (c/sqrt(c 2 +8ab)) >=3{ abc/(3 3/2 4abc)} 1/3 or ((a/sqrt(a 2 +8bc)) + (b/sqrt(b 2 +8ac)) + (c/sqrt(c 2 +8ab)) >= &radic;3 /4 1/3 >=1.09 >1 Please feel free to post as many doubts on our discussion forum as you can. If you find any question Difficult to understand - post it here and we will get you the answer and detailed solution very quickly. We are all IITians and here to help you in your IIT JEE preparation. All the best. Regards, Askiitians Experts Badiuddin

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How does one prove that (a/sqrt(a 2 +8bc)) + (b/sqrt(b 2 +8ac)) + (c/sqrt(c 2 +8ab)) >= 1 ? sqrt() is the square root function

How does one prove that

(a/sqrt(a²+8bc)) + (b/sqrt(b²+8ac)) + (c/sqrt(c²+8ab)) >= 1 ?

sqrt() is the square root function

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How does one prove that (a/sqrt(a 2 +8bc)) + (b/sqrt(b 2 +8ac)) + (c/sqrt(c 2 +8ab)) >= 1 ? sqrt() is the square root function

How does one prove that (a/sqrt(a2+8bc)) + (b/sqrt(b2+8ac)) + (c/sqrt(c2+8ab)) >= 1 ? sqrt() is the square root function

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How does one prove that

(a/sqrt(a²+8bc)) + (b/sqrt(b²+8ac)) + (c/sqrt(c²+8ab)) >= 1 ?

sqrt() is the square root function