Question

已知a,b∈R,f(x)=x^2-abx.证明不等式f(a^2)+f(b^2)≥0

Answer 1

f(a^2)+f(b^2)

=a^4-a^3b+b^4-ab^3

=a^3(a-b)-b^3(a-b)

=(a-b)(a^3-b^3)

=(a-b)^2(a^2+ab+b^2)

=(a-b)^2[(a+b/2)^2+3b^2/4]

≥0

当且仅当"a=b"时取"="