Question

Tìm GTNN của các biểu thức:

a) A = (x + 8)⁴ + (x + 6)4

b) B = (0,5x² + x)² - 3 |0,5x² + x|

c) C = (x - 1)(x + 3)(x² - 4x + 5)

d) D = x⁴ - 2x³ + 3x² - 2x + 1

e) E = x⁴ - 6x³ + 10x² - 6x + 9

g) G = (x² + x - 6) (x² + x + 2)

Answer 1

b)Ta có:\(B=\left(0,5x^2+x\right)^2-3\left|0,5x^2+x\right|\)

\(B=\left|0,5x^2+x\right|^2-3\left|0,5x^2+x\right|+\dfrac{9}{4}-\dfrac{9}{4}\)

\(B=\left(\left|0,5x^2+x\right|-\dfrac{3}{2}\right)^2-\dfrac{9}{4}\ge-\dfrac{9}{4}\)

"="<=>\(\left|0,5x^2+x\right|=\dfrac{3}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-3\end{matrix}\right.\)

g)Ta có:\(G=\left(x^2+x-6\right)\left(x^2+x+2\right)\)

Đặt \(x^2+x-2=t\)

\(\Rightarrow G=\left(t-4\right)\left(t+4\right)\)

\(G=t^2-16\ge-16\)

"="<=>\(x^2+x-2=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-2\end{matrix}\right.\)

Answer 2

E=\(x^4-6x^3+9x^2+x^2-6x+9\)

\(=x^2\left(x^2-6x+9\right)+x^2-6x+9\\ =x^2\left(x-3\right)^2+\left(x-3\right)^2\ge0\forall x\\ E_{min}=0\Leftrightarrow x=3\)