a, A= $\frac{x}{2}$+$\frac{18}{x-1}$
= $\frac{x-1}{2}$+$\frac{18}{x-1}$+$\frac{1}{2}$
Áp dụng bđt cô si, ta có: A≥ 2.$\sqrt[]{\frac{x-1}{2}.\frac{18}{x-1}}$+$\frac{1}{2}$ = 2.3+$\frac{1}{2}$ = $\frac{14}{5}$
⇒ A≥ $\frac{14}{5}$
Dấu = xảy ra khi $\frac{x-1}{2}$= $\frac{18}{x-1}$
⇔ ( x-1)²= 36
⇔ x-1= ±6
⇔ x= 7 ( tm)
hoặc x= -5 ( ktm)
Vậy Amin= $\frac{14}{5}$ khi x= 7
b, B= $\frac{x}{3}$+$\frac{5}{2x-1}$
= $\frac{2x-1}{6}$+$\frac{5}{2x-1}$+$\frac{1}{6}$
Tiếp tục áp dụng bđt cô si, ta có: B≥ 2.$\sqrt[]{\frac{5}{6}}$+$\frac{1}{6}$
Dấu = xảy ra khi $\frac{2x-1}{6}$= $\frac{5}{2x-1}$
⇔ ( 2x-1)²= 30
⇔ 2x-1= ± $\sqrt[]{30}$
⇔ \(\left[ \begin{array}{l}x=\frac{\sqrt[]{30}+1}{2}( tm)\\x=\frac{-\sqrt[]{30}+1}{2}(ktm)\end{array} \right.\)
Vậy Bmin= 2.$\sqrt[]{\frac{5}{6}}$+$\frac{1}{6}$ khi x= $\frac{\sqrt[]{30}+1}{2}$
