a) \(P=y^2+8y+15\)
\(P=y^2+2.y.4+16-1\)
\(P=\left(y+4\right)^2-1\)
Vì \(\left(y+4\right)^2\ge0\) với mọi y
\(\Rightarrow\left(y+4\right)^2-1\ge-1\) với mọi y
\(\Rightarrow Pmin=-1\Leftrightarrow y=-4\)
b) \(A=u^2+v^2-2u+3v+15\)
\(A=u^2-2u+1+v^2+2.v.\dfrac{3}{2}+\dfrac{9}{4}-\dfrac{9}{4}+14\)
\(A=\left(u-1\right)^2+\left(v+\dfrac{3}{2}\right)^2+\dfrac{47}{4}\)
Vì \(\left(u-1\right)^2\ge0\) với mọi u
\(\left(v+\dfrac{3}{2}\right)^2\ge0\) với mọi v
\(\Rightarrow\left(u-1\right)^2+\left(v+\dfrac{3}{2}\right)^2\ge0\) với mọi u,v
\(\Rightarrow\left(u-1\right)^2+\left(v+\dfrac{3}{2}\right)^2+\dfrac{47}{4}\ge\dfrac{47}{4}\)
\(\Rightarrow Amin=\dfrac{47}{4}\Leftrightarrow\left\{{}\begin{matrix}u=1\\v=-\dfrac{3}{2}\end{matrix}\right.\)