Với x>9 tìm gtnn của 4x/(√x-3)

Đáp án:

\(Min\,\,\,P = 48\,\,khi\,\,\,x = 36\)

Giải thích các bước giải:

\(\begin{array}{l} x > 9\\ P = \frac{{4x}}{{\sqrt x - 3}} = \frac{{4x - 36 + 36}}{{\sqrt x - 3}}\\ = \frac{{4\left( {\sqrt x - 3} \right)\left( {\sqrt x + 3} \right)}}{{\sqrt x - 3}} + \frac{{36}}{{\sqrt x - 3}}\\ = 4\left( {\sqrt x + 3} \right) + \frac{{36}}{{\sqrt x - 3}}\\ = 4\left( {\sqrt x - 3} \right) + \frac{{36}}{{\sqrt x - 3}} + 24\\ Ap\,\,dung\,\,bdt\,\,Co - si\,\,\,cho\,\,hai\,\,so\,\,duong:\,\,\,4\left( {\sqrt x - 3} \right)\,\,\,va\,\,\,\frac{{36}}{{\sqrt x - 3}}\,\,\,ta\,\,co:\\ 4\left( {\sqrt x - 3} \right) + \frac{{36}}{{\sqrt x - 3}} \ge 2\sqrt {4\left( {\sqrt x - 3} \right).\frac{{36}}{{\sqrt x - 3}}} = 24\\ \Rightarrow P \ge 24 + 24 = 48.\\ Dau\,\,\, = \,\,xay\,\,ra \Leftrightarrow 4\left( {\sqrt x - 3} \right) = \frac{{36}}{{\sqrt x - 3}}\\ \Leftrightarrow 4{\left( {\sqrt x - 3} \right)^2} = 36\\ \Leftrightarrow {\left( {\sqrt x - 3} \right)^2} = 9\\ \Leftrightarrow \sqrt x - 3 = 3\,\,\,\left( {do\,\,x > 9 \Rightarrow \sqrt x - 3 > 0} \right)\\ \Leftrightarrow \sqrt x = 6\\ \Leftrightarrow x = 36\,\,\left( {tm} \right)\\ Vay\,\,Min\,\,\,P = 48\,\,khi\,\,\,x = 36. \end{array}\)