Question

1. Tìm giá trị nhỏ nhất của:

a) A = x² - x - 2

b) B = x² - 3x + 4

2. Tìm giá trị lớn nhất của:

a) A = -x² + x

b) B = -x² + 2x + 2

c) C = -4x² - 4x + 5

d) D = -9x² - 24x + 1

Answer 1

\(x^2-x-2=x^2-x+\frac{1}{4}-\frac{9}{4}=\left(x-\frac{1}{2}\right)^2-\frac{9}{4}\ge0-\frac{9}{4}=\frac{-9}{4}\Rightarrow A_{min}=\frac{-9}{4}.\)

Dâu "=" xay ra \(\Leftrightarrow x+\frac{1}{2}=0\Leftrightarrow x=\frac{-1}{2}\)

\(x^2-3x+4=x^2-3x+\frac{9}{4}+\frac{7}{4}=x^2-2.\frac{3}{2}x+\frac{9}{4}+\frac{7}{4}=\left(x+\frac{3}{2}\right)^2+\frac{7}{4}\ge0+\frac{7}{4}=\frac{7}{4}\Rightarrow B_{min}=\frac{7}{4}\)

Dâu "=" xay ra \(\Leftrightarrow x+\frac{3}{2}=0\Leftrightarrow x=\frac{-3}{2}\)

Answer 2

Bài 2:

a) \(A=-\left(x^2-x\right)=-\left(x^2-2.x.\frac{1}{2}+\frac{1}{4}-\frac{1}{4}\right)=-\left(x-\frac{1}{2}\right)^2+\frac{1}{4}\le\frac{1}{4}\)

Đẳng thức xảy ra khi x = 1/2

b) \(B=-\left(x^2-2x-2\right)=-\left(x^2-2x+1-3\right)\)

\(=-\left(x-1\right)^2+3\le3\)

"=" <=> x = 1

c) \(C=-4\left(x^2+x-\frac{5}{4}\right)=-4\left(x^2+2.x.\frac{1}{2}+\frac{1}{4}-\frac{6}{4}\right)\)

\(=-4\left(x+\frac{1}{2}\right)^2+6\le6\)

"=" <=> x = -1/2

d) \(D=-9\left(x^2+\frac{24}{9}.x-\frac{1}{9}\right)=-9\left(x^2+2.x.\frac{4}{3}+\frac{16}{9}-\frac{16}{9}-\frac{1}{9}\right)\)

\(=-9\left(x+\frac{4}{3}\right)^2+17\le17\)

Đẳng thức xảy ra khi x = -4/3