Question

Tìm GTLN của biểu thức:

a) \(A=-2x+\sqrt{x}\)

b) \(B=-x+5\sqrt{x}\)

c) \(C=-x+1+2\sqrt{x-1}\)

Answer 1

\(a.A=-2x+\sqrt{x}=-2\left(x-2.\dfrac{1}{4}\sqrt{x}+\dfrac{1}{16}\right)+\dfrac{1}{8}=-2\left(\sqrt{x}-\dfrac{1}{4}\right)^2+\dfrac{1}{8}\le\dfrac{1}{8}\left(x\ge0\right)\)

\(\Rightarrow A_{Max}=\dfrac{1}{8}."="\Leftrightarrow x=\dfrac{1}{16}\left(TM\right)\)

\(b.B=-x+5\sqrt{x}=-\left(x-2.\dfrac{5}{2}\sqrt{x}+\dfrac{25}{4}\right)+\dfrac{25}{4}=-\left(\sqrt{x}-\dfrac{5}{2}\right)^2+\dfrac{25}{4}\le\dfrac{25}{4}\left(x\ge0\right)\)

\(\Rightarrow B_{Max}=\dfrac{25}{4}."="\Leftrightarrow x=\dfrac{25}{4}\left(TM\right)\)

\(c.C=-x+1+2\sqrt{x-1}=-\left(x-1-2\sqrt{x-1}+1\right)+1=-\left(\sqrt{x-1}-1\right)^2+1\le1\left(x\ge1\right)\)

\(\Rightarrow C_{Max}=1."="\Leftrightarrow x=2\left(TM\right)\)