Question

[Nguyễn Thị Ngọc Hân]

Tìm GTLN và GTNN của \(P=\frac{3x^2-4x+8}{x^2+2}\)

Answer 1

[𝓓𝓾𝔂 𝓐𝓷𝓱]

1) Tìm GTNN:

\(P=\frac{3x^2-4x+8}{x^2+2}=\frac{2\left(x^2+2\right)+x^2-4x+4}{x^2+2}=2+\frac{\left(x-2\right)^2}{x^2+2}\ge2\)

Dấu "=" xảy ra \(\Leftrightarrow x-2=0\Leftrightarrow x=2\)

Vậy \(P_{Min}=2\) khi \(x=2\)

2) Tìm GTLN:

\(P=\frac{3x^2-4x+8}{x^2+2}=\frac{5\left(x^2+2\right)-2x^2-4x-2}{x^2+2}=5-\frac{2\left(x+1\right)^2}{x^2+2}\le5\)

Dấu "=" xảy ra \(\Leftrightarrow x+1=0\Leftrightarrow x=-1\)

Vậy \(P_{Max}=5\) khi \(x=-1\)

Answer 2

[Nguyễn Việt Lâm]

\(\Leftrightarrow Px^2+2P=3x^2-4x+8\)

\(\Leftrightarrow\left(P-3\right)x^2+4x+2P-8=0\)

\(\Delta'=4-\left(P-3\right)\left(2P-8\right)\ge0\)

\(\Rightarrow-P^2+7P-10\ge0\)

\(\Rightarrow2\le P\le5\)

\(P_{min}=2\) khi \(x=2\)

\(P_{max}=5\) khi \(x=-1\)