\(A=x^2-4x+1\)
\(A=\left(x^2-4x+4\right)-3\)
\(A=\left(x-2\right)^2-3\)
Vì \(\left(x-2\right)^2\ge0\) với mọi x
\(\Rightarrow\left(x-2\right)^2-3\ge-3\) với mọi x
\(\Rightarrow Amin=-3\Leftrightarrow x=2\)
\(B=4x^2+4x+11\)
\(B=\left(4x^2+4x+1\right)+10\)
\(B=\left(2x+1\right)^2+10\)
Vì \(\left(2x+1\right)^2\ge0\) với mọi x
\(\Rightarrow\left(2x+1\right)^2+10\ge10\) với mọi x
\(\Rightarrow Bmin=10\Leftrightarrow x=-\dfrac{1}{2}\)
\(C=\left(x-1\right)\left(x+3\right)\left(x+2\right)\left(x+6\right)\)
\(C=\left[\left(x-1\right)\left(x+6\right)\right]\left[\left(x+2\right)\left(x+3\right)\right]\)
\(C=\left(x^2+5x-6\right)\left(x^2+5x+6\right)\)
\(C=\left(x^2+5x\right)^2-36\)
Vì \(\left(x^2+5x\right)^2\ge0\) với mọi x
\(\Rightarrow\left(x^2+5x\right)^2-36\ge-36\)
\(\Rightarrow Cmin=-36\Leftrightarrow x^2+5x=0\)
\(\Rightarrow x\left(x+5\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}x=0\\x=-5\end{matrix}\right.\)
\(D=5-8x-x^2\)
\(D=-\left(x^2+8x-5\right)\)
\(D=-\left(x^2+8x+16-16-5\right)\)
\(D=-\left(x^2+8x+16\right)+21\)
\(D=-\left(x+4\right)^2+21\)
Vì \(-\left(x+4\right)^2\le0\) với mọi x
\(\Rightarrow-\left(x+4\right)^2+21\le21\) với mọi x
\(\Rightarrow Dmax=21\Leftrightarrow x=-4\)
\(E=4x-x^2+1\)
\(E=-\left(x^2-4x-1\right)\)
\(E=-\left(x^2-4x+4-4-1\right)\)
\(E=-\left(x^2-4x+4\right)+5\)
\(E=-\left(x-2\right)^2+5\)
Vì \(-\left(x-2\right)^2\le0\) với mọi x
\(\Rightarrow-\left(x-2\right)^2+5\le5\) với mọi x
\(\Rightarrow Emax=5\Leftrightarrow x=2\)
