Prove these inequalities?

a) a^4 + b^4 >/= a^3b + ab^3

b) a^2 + b^2 + c^2 >/= ab + bc + ac


ans:
a)
we have following from AM GM enaquality

a^4 + (ab)^2 >= 2a^3.b .. 1
b^4 + (ab)^2 >= 2b^3.a ... 2
a^4 + b^4 >= 2(ab)^2 .. 3

add to get 2(a^4+b^4 ) + 2a^2b^2 >= 2a^3b + 2 ab^3 + 2 a^2 b^2
or (a^4+b^4 ) >= a^3b + ab^3

hence proved
b) we know a^2 + b^2 > = 2ab (from AM GM enaquality)
b^2 + c^2 >= 2bc
c^2 + a^2 >= 2ac

add to get 2(a^2 + b^2 + c^2) >= 2(ab+bc+ca)
or (a^2 + b^2 + c^2) >= ab + bc + ca