Ta có: \(E=5+\left(1-x\right)\left(x+2\right)\left(x+3\right)\left(x+6\right)\)
\(=5-\left(x-1\right)\left(x+2\right)\left(x+3\right)\left(x+6\right)\)
\(=5-\left[\left(x-1\right)\left(x+6\right)\right]\left[\left(x+2\right)\left(x+3\right)\right]\)
\(=5-\left(x^2+5x-6\right)\left(x^2+5x-6\right)\)
Đặt \(t=x^2+6x\)
\(\Rightarrow E=5+\left(t-6\right)\left(t+6\right)\)
\(=5+t^2-36\)
\(=t^2-31\)
Mà \(t^2\ge0\Rightarrow t^2-31\ge-31\)
\(\Rightarrow E\ge-31\)
Dấu "=" xảy ra \(\Leftrightarrow t^2=0\Leftrightarrow t=0\Leftrightarrow x^2+6x=0\Leftrightarrow x\left(x+6\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=-6\end{matrix}\right.\)
\(E=5+\left(1-x\right)\left(x+2\right)\left(x+3\right)\left(x+6\right)\\ E=5-\left[\left(x-1\right)\left(x+6\right)\right]\left[\left(x+2\right)\left(x+3\right)\right]\\ E=5-\left(x^2+5x-6\right)\left(x^2+5x+6\right)\)
Cách 1: \(E=5-\left(x^2+5x\right)^2+36=-\left(x^2+5x\right)^2+41\le41\)
\(E_{max}=41\Leftrightarrow x^2+5x=0\Leftrightarrow\left[{}\begin{matrix}x=-5\\x=0\end{matrix}\right.\)
Cách 2: Đặt \(x^2+5x=t\)
\(\Leftrightarrow E=5-\left(t+6\right)\left(t-6\right)=5-t^2+36=-t^2+41\le41\\ E_{max}=41\Leftrightarrow t=0\Leftrightarrow x^2+5x=0\Leftrightarrow\left[{}\begin{matrix}x=0\\x=-5\end{matrix}\right.\)