Question

Tìm giá trị nhỏ nhất, lớn nhất nếu:

\(K=x^2-7x+13\)

\(M=-x^2+4x-14\)

\(D=2x^2+4y^2+4xy+2x+4y+9\)

Answer 1

\(K=x^2-7x+13=x^2-2.x.\dfrac{7}{2}+\dfrac{49}{4}+\dfrac{3}{4}=\left(x-\dfrac{7}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\)

\(\Rightarrow Min_K=\dfrac{3}{4}\Leftrightarrow x=\dfrac{7}{2}\)

\(M=-x^2+4x-14=-\left(x^2-4x+4+10\right)=-\left(x-2\right)^2-10\le-10\)

\(\Rightarrow Max_M=-10\Leftrightarrow x=2\)

\(D=2x^2+4y^2+4xy+2x+4y+9\)

\(D=x^2+4xy+4y^2+2x+4y+x^2+1+8\)

\(D=\left(x+2y\right)^2+2\left(x+2y\right)+1+x^2+8\)

\(D=\left(x+2y+1\right)^2+x^2+8\ge8\)

\(\Rightarrow Min_D=8\Leftrightarrow x=0;y=-\dfrac{1}{2}\)