Question

[Nguyễn Hoàng Anh]

Tìm giá trị lớn nhất của biểu thức P=\(\dfrac{4x-1}{x^2+3}\)

Answer 1

[Lấp La Lấp Lánh]

Khi \(x=\dfrac{1}{4}\Leftrightarrow P=\dfrac{4.\dfrac{1}{4}-1}{\left(\dfrac{1}{4}\right)^2+3}=0\)

Khi \(x\ne\dfrac{1}{4}\Leftrightarrow P\ne\dfrac{4.\dfrac{1}{4}-1}{\left(\dfrac{1}{4}\right)^2+3}\Leftrightarrow P\ne0\)

\(P=\dfrac{4x-1}{x^2+3}\Leftrightarrow Px^2-4x+3P+1=0\) là pt bậc 2 do \(P\ne0\)

\(\Delta'=\left(-2\right)^2-P\left(3P+1\right)=-3P^2-P+4\)

Để pt có nghiệm thì \(\Delta'\ge0\Leftrightarrow-3P^2-P+4\ge0\Leftrightarrow-3\left(P+\dfrac{1}{6}\right)^2+\dfrac{49}{12}\ge0\Leftrightarrow P\le1\)

\(maxP=1\Leftrightarrow\dfrac{4x-1}{x^2+3}=1\Leftrightarrow x^2-4x+4=0\Leftrightarrow x=2\left(tm\right)\)

Answer 2

[Hồng Phúc]

\(P=\dfrac{4x-1}{x^2+3}\)

\(\Leftrightarrow x^2P+3P-4x+1=0\)

\(\Leftrightarrow Px^2-4x+3P+1=0\left(1\right)\)

\(\left(1\right)\) có nghiệm khi:

\(\Delta'=4-P\left(3P+1\right)=-3P^2-P+4\ge0\)

\(\Leftrightarrow P\in\left[-\dfrac{4}{3};1\right]\)

\(\Rightarrow P_{max}=1\Leftrightarrow x=2\)