Question

Cho x,y,z=0 thỏa mãn x^2+y^2+z^2=1 Tìm GTNN của M=1/16x^2+1/4y^2+1/z^2

Answer 1

x,y,z>0 nha

Answer 2

\(M=\dfrac{1}{16x^2}+\dfrac{1}{4y^2}+\dfrac{1}{16z^2}=\dfrac{1}{16}\left(\dfrac{1}{x^2}+\dfrac{2^2}{y^2}+\dfrac{4^2}{z^2}\right)\)

\(\Rightarrow M\ge\dfrac{1}{16}.\dfrac{\left(1+2+4\right)^2}{\left(x^2+y^2+z^2\right)}=\dfrac{49}{16}\)

\(\Rightarrow M_{min}=\dfrac{49}{16}\) khi \(\dfrac{1}{x^2}=\dfrac{2}{y^2}=\dfrac{4}{z^2}\Rightarrow\left\{{}\begin{matrix}x^2=\dfrac{1}{7}\\y^2=\dfrac{2}{7}\\z^2=\dfrac{4}{7}\end{matrix}\right.\)