Question

1)Cho 2 số x,y thoả man x+y=2

Cm x^4

+y^4>=2

Answer 1

Ta có:\(\left(x^2-y^2\right)^2\ge0\)

\(\Rightarrow x^4-2x^2y^2+y^4\ge0\)

\(\Rightarrow x^4+y^4\ge2x^2y^2\)

\(\Rightarrow x^4+y^4+x^4+y^4\ge x^4+2x^2y^2+y^4\)

\(\Rightarrow2\left(x^4+y^4\right)\ge\left(x^2+y^2\right)^2\)

\(\Rightarrow x^4+y^4\ge\dfrac{\left(x^2+y^2\right)^2}{2}\)

CMTT\(\Rightarrow x^2+y^2\ge\dfrac{\left(x+y\right)^2}{2}=\dfrac{2^2}{2}=2\)

\(\Rightarrow x^4+y^4\ge\dfrac{\left(x^2+y^2\right)^2}{2}\ge\dfrac{2^2}{2}=2\)

\(\Rightarrowđpcm\)