Question

Bài 1. Tìm GTNN

\(A=\left(x^2+x+1\right)^2\)

\(B=x^4-6x^3+10x^2-6x+9\)

Bài 2. Tìm GTLN:

\(M=\frac{3}{4x^2-4x+5}\)

Answer 1

\(A=\left(x^2+x+1\right)^2=\left[\left(x^2+x+\frac{1}{4}\right)+\frac{3}{4}\right]^2=\left[\left(x+\frac{1}{2}\right)^2+\frac{3}{4}\right]^2\ge\frac{9}{16}\)\("="\Leftrightarrow x=-\frac{1}{2}\)

\(B=x^4-6x^3+10x^2-6x+9\)

\(B=\left(x^4-6x^3+9x^2\right)+\left(x^2-6x+9\right)\)

\(B=x^2\left(x^2-6x+9\right)+\left(x^2-6x+9\right)=\left(x^2+1\right)\left(x-3\right)^2\ge0\)\("="\Leftrightarrow x=3\)

\(M=\frac{3}{4x^2-4x+5}=\frac{3}{4x^2-4x+1+4}=\frac{3}{\left(2x-1\right)^2+4}\le\frac{3}{4}\)

\("="\Leftrightarrow x=\frac{1}{2}\)