Question

Tìm GTLN:
a,11-10x-\(x^2\)
b,|x-4|(2-|x-4|)

Answer 1

a)Đặt  \(A=11-10x-x^2=-\left(x^2+10x+25\right)+36=-\left(x+5\right)^2+36\le36\)

Dấu = xảy ra \(\Leftrightarrow x+5=0\Leftrightarrow x=-5\)

Vậy GTLN của A là 36 \(\Leftrightarrow x=-5\)

Answer 2

Đặt \(B=\left|x-4\right|\left(2-\left|x-4\right|\right)\)

TH1:\(x\ge4\)

Suy ra \(B=\left(x-4\right)\left(2-x+4\right)=\left(x-4\right)\left(6-x\right)=-x^2+10x-24=-\left(x^2-10x+25\right)+1=-\left(x-5\right)^2+1\le1\)

Dấu = xảy ra \(\Leftrightarrow x-5=0\Leftrightarrow x=5\)

TH2: \(x< 4\)

Suy ra \(B=\left(4-x\right)\left(2-4+x\right)=\left(4-x\right)\left(x-2\right)=-x^2+6x-8=-\left(x^2-6x+9\right)+1=-\left(x-3\right)^2+1\le1\)

Dấu = xảy ra \(\Leftrightarrow x-3=0\Leftrightarrow x=3\)

Vậy GTLN của B là 1 \(\Leftrightarrow\)\(\left[\begin{array}{nghiempt}x=5\\x=3\end{array}\right.\)