Question

1, Tìm giá trị nhỏ nhất

a, A=x²+3x-13

b, B=5x²-15x-4

2,Tìm giá trị lớn nhất

a, C=2x-x²+1

b, D=-3x²+8x-10

Answer 1

Bài 1:

a) \(A=x^2+3x-13\)

\(A=x^2+2\cdot x\cdot\frac{3}{2}+\frac{9}{4}-\frac{61}{4}\)

\(A=\left(x+\frac{3}{2}\right)^2-\frac{61}{4}\ge-\frac{61}{4}\forall x\)

Dấu "=" \(\Leftrightarrow x=\frac{-3}{2}\)

b) \(B=5x^2-15x-4\)

\(B=5\left(x^2-3x-\frac{4}{5}\right)\)

\(B=5\left(x^2-2\cdot x\cdot\frac{3}{2}+\frac{9}{4}-\frac{61}{20}\right)\)

\(B=5\left[\left(x-\frac{3}{2}\right)^2-\frac{61}{20}\right]\)

\(B=5\left(x-\frac{3}{2}\right)^2-\frac{61}{4}\ge-\frac{61}{4}\forall x\)

Dấu "=" \(\Leftrightarrow x=\frac{3}{2}\)

Bài 2:

a) \(C=2x-x^2+1\)

\(C=-\left(x^2-2x-1\right)\)

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

\(C=-\left[\left(x-1\right)^2-2\right]\)

\(C=2-\left(x-1\right)^2\le2\forall x\)

Dấu "=" \(\Leftrightarrow x=1\)

b) \(D=-3x^2+8x-10\)

\(D=-3\left(x^2-\frac{8}{3}x+\frac{10}{3}\right)\)

\(D=-3\left(x^2-2\cdot x\cdot\frac{4}{3}+\frac{16}{9}+\frac{14}{9}\right)\)

\(D=-3\left[\left(x-\frac{4}{3}\right)^2+\frac{14}{9}\right]\)

\(D=\frac{-14}{3}-3\left(x-\frac{4}{3}\right)^2\le\frac{-14}{3}\forall x\)

Dấu "=" \(\Leftrightarrow x=\frac{4}{3}\)