Question

Tìm giá trị lớn nhất M của hàm số \(f\left(x\right)=\sqrt{x-1}+\sqrt{3-x}-2\sqrt{-x^2+4x-3}\) 

A. M=0 B. M=\(-\sqrt{2}\) C.M= \(\sqrt{2}\) D. M=9/4

Answer 1

Chọn B

Answer 2

ĐKXĐ: \(1\le x\le3\)

Đặt \(\sqrt{x-1}+\sqrt{3-x}=t\)

Ta có: \(\sqrt{x-1}+\sqrt{3-x}\ge\sqrt[]{x-1+3-x}=\sqrt{2}\)

\(\sqrt{x-1}+\sqrt{3-x}\le\sqrt{2\left(x-1+3-x\right)}=2\)

\(\Rightarrow t\in\left[\sqrt{2};2\right]\)

\(t^2=x-1+3-x+2\sqrt{\left(x-1\right)\left(3-x\right)}=2+2\sqrt{-x^2+4x-3}\)

\(\Rightarrow-2\sqrt{-x^2+4x-3}=2-t^2\)

\(\Rightarrow f\left(x\right)=f\left(t\right)=t+2-t^2\)

\(f'\left(t\right)=1-2t=0\Rightarrow t=\dfrac{1}{2}\notin\left[\sqrt{2};2\right]\)

\(f\left(\sqrt{2}\right)=\sqrt{2}\) ; \(f\left(2\right)=0\)

\(\Rightarrow M=\sqrt{2}\)