If largest constant such that Kabc a + b + c (a+b)^2+(a+b+4 c)^2 a, b, c 0 is k then K 25 is equal to _______.

Step-by-step solution

Using AM - GM inequality aligned & (a+b)^2+(a+b+4 c)^2=(a+b)^2+(a+2 c+b+2 c)^2 (2 a b )^2+(2 2 a c +2 2 b c )^2 & =4 a b+8 a c+8 b c+16 c a b & (a+b)^2+(a+b+4 c)^2 a b c (a+b+c) 4 a b+8 a c+8 b c+16 c a b a b c (a+b+c) & = [ 4 c + 8 b + 8 a + 16 a b ](a+b+c) [ a 2 + a 2 + b 2 + b 2 +c ] 8 [5 [5] 1 2 a^2 b^2 c ] (5 [5] a^2 b^2 c 2^4 ]=100 aligned