\(M=x^2-4x+1=x^2-4x+4-3=\left(x-2\right)^2-3\) Do \(\left(x-2\right)^2\ge0=>\left(x-2\right)^2-3\ge-3\)
Vậy min M=-3 khi x=2
\(N=x^2+10x+50=x^2+10x+25+25=\left(x+5\right)^2+25.\) Do \(\left(x+5\right)^2\ge0\Rightarrow\left(x+5\right)^2+25\ge25\Rightarrow N_{min}=25\) khi x=-5
\(P=x^2+12x-1=x^2+12x+36-37=\left(x+6\right)^2-37\) Do \(\left(x+6\right)^2\ge0\Rightarrow\left(x+6\right)^2-37\ge-37\Rightarrow P_{min}=-37\) khi x=-6
\(Q=x^2-x+1=\left(x^2-x+\frac{1}{4}\right)+\frac{3}{4}=\left(x-\frac{1}{2}\right)^2+\frac{3}{4}\) Do \(\left(x-\frac{1}{2}\right)^2\ge0\Rightarrow\left(x-\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\Rightarrow Q_{min}=\frac{3}{4}\) khi \(x=\frac{1}{2}\)
\(R=x^2-3x+2=\left(x^2-3x+2,25\right)-0,5=\left(x-1,5\right)^2-0,5\) Do \(\left(x-1,5\right)^2\ge0\Rightarrow\left(x-1,5\right)^2-0,5\ge-0,5\Rightarrow R_{min}=-0,5\) khi x=1,5
\(S=2x^2-8x+1=2\left(x^2-4x+2\right)-3=2\left(x-2\right)^2-3\) Do \(2\left(x-2\right)^2\ge0\Rightarrow2\left(x-2\right)^2-3\ge-3\Rightarrow S_{min}=-3\) khi x=2
\(T=2x^2+6x+1=2\left(x^2+3x+2,25\right)-3,5=2\left(x+1,5\right)^2-3,5\) Do \(2\left(x+1,5\right)^2\ge0\Rightarrow2\left(x+1,5\right)^2-3,5\ge-3,5\Rightarrow T_{min}=-3,5\) khi x=-1,5
\(V=3x^2+x+2=3\left(x^2+\frac{1}{3}x+\frac{1}{36}\right)+\frac{23}{24}=3\left(x+\frac{1}{6}\right)^2+\frac{23}{24}\) Do\(3\left(x+\frac{1}{6}\right)^2\ge0\Rightarrow3\left(x+\frac{1}{6}\right)^2+\frac{23}{24}\ge\frac{23}{24}\Rightarrow V_{min}=\frac{23}{24}\) khi \(x=\frac{1}{6}\)
