Question

+) Cho \(x,y,z\in\left[-1;1\right]\) và \(x+y+z=0\)

\(CMR:\sqrt{1+x+y^2}+\sqrt{1+y+z^2}+\sqrt{1+z+x^2}\ge3\)

+) Cho \(\left\{{}\begin{matrix}a,b,c>0\\a^4+b^4+c^4=3\end{matrix}\right.\)

\(CMR:ab^2+bc^2+c+a^2\le4\)

Answer 1

Bài 2:

\(ab^2\le\dfrac{a^2+b^4}{2}\);   \(bc^2\le\dfrac{b^2+c^4}{2}\);   \(c\le\dfrac{c^2+1}{2}\);   \(a^2\le\dfrac{a^4+1}{2}\)

Cộng theo vế các BĐT trên, ta có:

\(VT\le\dfrac{1}{2}\left(a^4+b^4+c^4\right)+\dfrac{1}{2}\left(a^2+b^2+c^2\right)+1\)

\(\le\dfrac{1}{2}\left(a^4+b^4+c^4\right)+\dfrac{1}{2}\sqrt{3\left(a^4+b^4+c^4\right)}+1\)

\(=\dfrac{3}{2}+\dfrac{3}{2}+1=4\)

Đẳng thức xảy ra\(\Leftrightarrow a=b=c=1\)