Squares
( a + 2 b a ) 2 + ( b + 2 c b ) 2 + ( c + 2 a c ) 2
If a , b and c are positive reals, find the minimum value of the expression above to 2 decimal places.
The answer is 0.33.
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2 solutions
Isn't the first approach Cauchy-Schwarz instead of T2 since we are using ( 1 + 1 + 1 ) ( ( a + 2 b a ) 2 + ( b + 2 c b ) 2 + ( c + 2 a c ) 2 ) ≥ ( a + 2 b a + b + 2 c b + c + 2 a c ) 2 instead of the engel form?
Nicely done, anyway. My approach is basically the same as yours.
a=b=c=1 for minima .nice way to go on with the problen @Gurīdo Cuong
Call the expression F Now using titu's lemma we get F ≥ 3 1 ( a + 2 b a + b + 2 c b + c + 2 a c ) 2 Using it again and we have a + 2 b a + b + 2 c b + c + 2 a c = a 2 + 2 a b a 2 + b 2 + 2 b c b 2 + c 2 + 2 a c c 2 ≥ 1 So F ≥ 3 1 , the equality holds when a = b = c