Question

tìm max \(A=\dfrac{x^4}{\left(x+1\right)^6}\left(x>0\right)\)

Answer 1

\(A=\left(\dfrac{x^2}{\left(x+1\right)^3}\right)^2=\left(\dfrac{x^2}{x^3+3x^2+3x+1}\right)^2=\left(\dfrac{1}{x+\dfrac{3}{x}+\dfrac{1}{x^2}+3}\right)^2\)

\(A=\left(\dfrac{1}{x+\dfrac{4}{x}+\left(\dfrac{1}{x}-\dfrac{1}{2}\right)^2+\dfrac{11}{4}}\right)^2\le\left(\dfrac{1}{x+\dfrac{4}{x}+\dfrac{11}{4}}\right)^2\)

\(A\le\left(\dfrac{1}{2\sqrt{\dfrac{4x}{x}}+\dfrac{11}{4}}\right)^2=\dfrac{16}{729}\)

Dấu "=" xảy ra khi \(x=2\)