Show that for real a , b , c, a 2 + b 2 +...

Frederick Kramer

Frederick Kramer

Answered

2022-07-12

Show that for real a , b , c, a 2 + b 2 + c 2 a b + b c + c a

Answer & Explanation

Zackery Harvey

Zackery Harvey

Expert

2022-07-13Added 21 answers

Let f ( a , b , c ) = a 2 + b 2 + c 2 a b b c c a. Then we have, f ( t a , t , b , t c ) = t 2 f ( a , b , c ). Hence, f is homogeneous of degree two. For, t 0, we have f ( a , b , c ) > 0 f ( t a , t b , t c ) > 0. Therefore, we may make various normalizations. For example, we may set, a = 1 , b = 1 + x , c = 1 + y and get x 2 + y 2 x y = ( x y 2 ) 2 + 3 y 2 4 > 0
malalawak44

malalawak44

Expert

2022-07-14Added 4 answers

Using caushy-Schwartz Inequality:
( a 2 + b 2 + c 2 ) ( b 2 + c 2 + a 2 ) ( a b + b c + c a ) 2
and equality hold when a b = b c = c a .

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