Question

tìm max :\(A=\frac{xy\sqrt{z-5}+xz\sqrt{y-4}+yz\sqrt{x-3}}{xyz}\)

 

Answer 1

Viết lại A = \(\frac{\text{ }\sqrt{z-5}}{z}+\frac{\sqrt{y-4}}{y}+\frac{\sqrt{x-3}}{x}\)

Ta có : \(\sqrt{5\left(z-5\right)}\le\frac{5+z-5}{2}=\frac{z}{2}\Rightarrow\sqrt{z-5}\le\frac{z}{2\sqrt{5}}\) => \(\frac{z-5}{z}\le\frac{1}{2\sqrt{5}}\)  

tương tự \(\sqrt{y-4}\le\frac{y}{4}\Rightarrow\frac{\sqrt{y-4}}{y}\le\frac{1}{4}\)  

            \(\frac{\sqrt{x-3}}{x}\le\frac{1}{2\sqrt{3}}\)

=> A \(\le\frac{1}{2\sqrt{5}}+\frac{1}{4}+\frac{1}{2\sqrt{3}}\)

Vậy GTLN .... tại x = 6 ; y = 8 ; z = 10