Question

Cho \(M=\frac{5}{9x^2-6x+\frac{9}{4}}\) Tìm GTLN

Answer 1

\(M=\frac{5}{9x^2-6x+\frac{9}{4}}=\frac{5}{\left(3x\right)^2-6x+1+\frac{5}{4}}=\frac{5}{\left(3x-1\right)^2+\frac{5}{4}}\)

Có \(\left(3x-1\right)^2\ge0\) vs mọi x

<=> \(\left(3x-1\right)^2+\frac{5}{4}\ge\frac{5}{4}\) vs mọi x

<=> \(\frac{5}{\left(3x-1\right)^2+\frac{5}{4}}\le\frac{5}{\frac{5}{4}}=4\) với mọi x

<=> M\(\le4\)

Dấu "=" xảy ra <=> \(3x-1=0\) <=> \(x=\frac{1}{3}\)

Vậy maxM=4 <=> x=\(\frac{1}{3}\)