Question

tìm giá trị nhỏ nhất
M=x⁴+1\(/ \) (x²+1)²

Answer 1

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

\(M=\frac{\left(x^2+1\right)^2}{2\left(x^2+1\right)^2}+\frac{\left(x^2-1\right)^2}{\left(x^2+1\right)^2}=\frac{1}{2}+\left(\frac{x^2-1}{x^2+1}\right)^2\ge\frac{1}{2}\)

\(M_{min}=\frac{1}{2}\) khi \(x^2-1=0\Rightarrow x=\pm1\)