Question

Tìm min của D=x²+2y² biết: x+2y=1

Tìm min của E=2x²+5y² biết: 4x-3y=7

Answer 1

\(x+2y=1\Rightarrow x=1-2y\)

\(\Rightarrow D=x^2+2y^2=\left(1-2y\right)^2+2y^2=6y^2-4y+1=6\left(y-\dfrac{1}{3}\right)^2+\dfrac{1}{3}\ge\dfrac{1}{3}\)

\(D_{min}=\dfrac{1}{3}\) khi \(y=\dfrac{1}{3}\Rightarrow x=\dfrac{1}{3}\)

b.

\(4x-3y=7\Rightarrow y=\dfrac{4x-7}{3}\)

\(E=2x^2+5y^2=2x^2+5\left(\dfrac{4x-7}{3}\right)^2=\dfrac{98}{9}\left(x-\dfrac{10}{7}\right)^2+5\ge5\)

\(E_{min}=5\) khi \(x=\dfrac{10}{7}\Rightarrow y=-\dfrac{3}{7}\)