+) \(P=2x^2-3x+10=2.\left(x^2-\dfrac{3}{2}x\right)+10=2.\left(x^2-2.\dfrac{3}{4}x+\dfrac{9}{16}\right)-\dfrac{9}{8}+10\)
\(=2.\left(x-\dfrac{3}{4}\right)^2+\dfrac{71}{8}\ge\dfrac{71}{8}\) hay \(P\ge\dfrac{71}{8}\)
Dấu ''='' xảy ra \(\Leftrightarrow2.\left(x-\dfrac{3}{4}\right)^2=0\Rightarrow\left(x-\dfrac{3}{4}\right)^2=0\Rightarrow x-\dfrac{3}{4}=0\Rightarrow x=\dfrac{3}{4}\)
Vậy \(minP=\dfrac{71}{8}\) khi \(x=\dfrac{3}{4}\)
+) \(Q=5x^2+2x-10=5.\left(x^2+\dfrac{2}{5}x\right)-10\)
\(=5.\left(x^2+2.\dfrac{1}{5}x+\dfrac{1}{25}\right)-\dfrac{1}{5}-10=5.\left(x+\dfrac{1}{5}\right)^2-\dfrac{51}{5}\ge-\dfrac{51}{5}\)
hay \(Q\ge-\dfrac{51}{5}\)
Dấu ''='' xảy ra \(\Leftrightarrow5.\left(x+\dfrac{1}{5}\right)^2=0\Rightarrow x+\dfrac{1}{5}=0\Rightarrow x=-\dfrac{1}{5}\)
Vậy \(minQ=-\dfrac{51}{5}\) khi \(x=-\dfrac{1}{5}\)
+) \(E=x^2-7x+100=x^2-2.x.\dfrac{7}{2}+\dfrac{49}{4}+\dfrac{351}{4}\)
\(=\left(x-\dfrac{7}{2}\right)^2+\dfrac{351}{4}\ge\dfrac{351}{4}\) hay \(E\ge\dfrac{351}{4}\)
Dấu ''='' xảy ra \(\Leftrightarrow\left(x-\dfrac{7}{2}\right)^2=0\Rightarrow x-\dfrac{7}{2}=0\Rightarrow x=\dfrac{7}{2}\)
Vậy \(minE=\dfrac{351}{4}\) khi \(x=\dfrac{7}{2}\)
+) \(F=4x^2+3x+20=4x^2+2.2x.\dfrac{3}{4}+\dfrac{9}{16}+\dfrac{311}{16}\)
\(=\left(2x+\dfrac{3}{4}\right)^2+\dfrac{311}{16}\ge\dfrac{311}{16}\) hay \(F\ge\dfrac{311}{16}\)
Dấu ''='' xảy ra \(\Leftrightarrow\left(2x+\dfrac{3}{4}\right)^2=0\Rightarrow2x+\dfrac{3}{4}=0\Rightarrow x=-\dfrac{3}{8}\)
Vậy \(minF=\dfrac{311}{16}\) khi \(x=\dfrac{-3}{8}\)
+) \(H=9x^2-5x+208=9x^2-2.3x.\dfrac{5}{6}+\dfrac{25}{36}+\dfrac{7463}{36}\)
\(=\left(3x-\dfrac{5}{6}\right)^2+\dfrac{7463}{36}\ge\dfrac{7463}{36}\) hay \(H\ge\dfrac{7463}{36}\)
Dấu ''='' xảy ra \(\Leftrightarrow\left(3x-\dfrac{5}{6}\right)^2=0\Rightarrow3x-\dfrac{5}{6}=0\Rightarrow x=\dfrac{5}{18}\)
Vậy \(minH=\dfrac{7463}{36}\) khi \(x=\dfrac{5}{18}\)
