Question

3)Cho a +b = 1.Tìm GTNN của A= a(a²+ 2b )+ b(b² -a)

Answer 1

a+b=1

=>a=1-b

\(A=a\left(a^2+2b\right)+b\left(b^2-a\right)\)

\(=a^3+2ab+b^3-ab\)

\(=a^3+b^3+ab=\left(a+b\right)^3-3ab\left(a+b\right)+ab\)

\(=1^3-3ab+ab=1-2ab\)

=1-2b(1-b)

\(=1-2b+2b^2=2\left(b^2-b+\frac12\right)=2\left(b^2-b+\frac14+\frac14\right)\)

\(=2\cdot\left(b-\frac12\right)^2+\frac12\ge\frac12\forall b\)

Dấu '=' xảy ra khi \(b-\frac12=0\)

=>\(b=\frac12\)

=>\(a=1-b=1-\frac12=\frac12\)