Question

Tìm GTNN hoặc GTLN của các biểu thức sau:

a) 2x² - x + 1

b) 5x - x² + 4

c) x² + 5y² - 2xy + 4y + 3

Answer 1

a) \(2x^2-x+1=2\left(x-\dfrac{1}{4}\right)^2+\dfrac{7}{8}\ge\dfrac{7}{8}\)

\(ĐTXR\Leftrightarrow x=\dfrac{1}{4}\)

b) \(5x-x^2+4=-\left(x-\dfrac{5}{2}\right)^2+\dfrac{41}{4}\le\dfrac{41}{4}\)

\(ĐTXR\Leftrightarrow x=\dfrac{5}{2}\)

c) \(x^2+5y^2-2xy+4y+3=\left(x-y\right)^2+\left(2y+1\right)^2+2\ge2\)

\(ĐTXR\Leftrightarrow\)\(x=y=-\dfrac{1}{2}\)

Answer 2

b: ta có: \(-x^2+5x+4\)

\(=-\left(x^2-5x-4\right)\)

\(=-\left(x^2-2\cdot x\cdot\dfrac{5}{2}+\dfrac{25}{4}-\dfrac{41}{4}\right)\)

\(=-\left(x-\dfrac{5}{2}\right)^2+\dfrac{41}{4}\le\dfrac{41}{4}\forall x\)

Dấu '=' xảy ra khi \(x=\dfrac{5}{2}\)

Answer 3