If `a ,b ,a n dc` are positive and `a+b+c=6,` show that `(a+1//b)2+(b+1//c)2+(c+1//a)2geq75//4.`

Explanation

`((1)/(a) + (1)/(b) + (1)/(c ))/(3) ge (3)/(a + b + c) = (3)/(6) = (1)/(2)`
so, `(1)/(a) + (1)/(b) + (1)/(c ) ge (3)/(2)`
Now, `((a + (1)/(b))^(2) + (b + (1)/(c ))^(2) + (c + (1)/(a))^(2))/(3) ge [((a + (1)/(b)) + (b + (1)/(c )) + (c + (1)/(a)))/(3)]^(2) ge [(6 + (3)/(2))/(3)]^(2) = (25)/(4)`
`:. (a + (1)/(b))^(2) + (b + (1)/(c ))^(2) + (c + (1)/(a))^(2) ge (75)/(4)`