Nếu \(x < 1\) thì \(x^{8} - x^{7} + x^{2} - x + 1\)

\(= x^{8} + x^{2} \left(\right. 1 - x^{5} \left.\right) + \left(\right. 1 - x \left.\right) > 0\).

Nếu \(x \geq 1\) thì \(x^{8} - x^{7} + x^{2} - x + 1\)

\(= x^{7} \left(\right. x - 1 \left.\right) + x \left(\right. x - 1 \left.\right) + 1 > 0\).

2(b2a2​+c2b2​+a2c2​)≥2(bc​+ab​+ca​)

Xét dấu hiệu \(2 \left(\right. \frac{a^{2}}{b^{2}} + \frac{b^{2}}{c^{2}} + \frac{c^{2}}{a^{2}} \left.\right) - 2 \left(\right. \frac{c}{b} + \frac{b}{a} + \frac{a}{c} \left.\right)\)

\(= \left(\right. \frac{a}{b} - \frac{b}{c} \left.\right)^{2} + \left(\right. \frac{b}{c} - \frac{c}{a} \left.\right)^{2} + \left(\right. \frac{c}{a} - \frac{a}{b} \left.\right)^{2} \geq 0\)

Nhân hai vế bất đẳng thức cần chứng minh với \(x + y\) ta được bất đẳng thức tương đương là

\(x^{5} + y^{5} > \left(\right. x^{2} + y^{2} \left.\right) \left(\right. x + y \left.\right)\) (1)

Từ giả thiết \(x > \sqrt{2}\) suy ra \(x^{2} > 2\) suy ra \(x^{5} > 2 x^{3}\), từ đó   

\(x^{5} + y^{5} > 2 \left(\right. x^{3} + y^{3} \left.\right)\)

\(= 2 \left(\right. x^{2} - x y + y^{2} \left.\right) \left(\right. x + y \left.\right)\)

\(= \left(\right. x - y \left.\right)^{2} + \left(\right. x^{2} + y^{2} \left.\right) \left(\right. x + y \left.\right) \geq \left(\right. x^{2} + y^{2} \left.\right) \left(\right. x + y \left.\right)\) suy ra (1), điều phải chứng minh.

x+y=1 nên \(\left(\right. 1 + \frac{1}{x} \left.\right) \left(\right. 1 + \frac{1}{y} \left.\right) - 9\)

\(= \frac{\left(\right. x + 1 \left.\right) \left(\right. y + 1 \left.\right) - 9 x y}{x y} = \frac{2 - 8 x y}{x y}\)  

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

\(= \frac{2 \left(\right. x - y \left.\right)^{2}}{x y} \geq 0\)

Dấu đẳng thức xảy ra khi và chỉ khi \(x = y = \frac{1}{2}\).

x+y=1 nên \(\left(\right. 1 + \frac{1}{x} \left.\right) \left(\right. 1 + \frac{1}{y} \left.\right) - 9\)

\(= \frac{\left(\right. x + 1 \left.\right) \left(\right. y + 1 \left.\right) - 9 x y}{x y} = \frac{2 - 8 x y}{x y}\)  

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

\(= \frac{2 \left(\right. x - y \left.\right)^{2}}{x y} \geq 0\)

Dấu đẳng thức xảy ra khi và chỉ khi \(x = y = \frac{1}{2}\).

x+y=1 nên \(\left(\right. 1 + \frac{1}{x} \left.\right) \left(\right. 1 + \frac{1}{y} \left.\right) - 9\)

\(= \frac{\left(\right. x + 1 \left.\right) \left(\right. y + 1 \left.\right) - 9 x y}{x y} = \frac{2 - 8 x y}{x y}\)  

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

\(= \frac{2 \left(\right. x - y \left.\right)^{2}}{x y} \geq 0\)

Dấu đẳng thức xảy ra khi và chỉ khi \(x = y = \frac{1}{2}\).

x+y=1 nên \(\left(\right. 1 + \frac{1}{x} \left.\right) \left(\right. 1 + \frac{1}{y} \left.\right) - 9\)

\(= \frac{\left(\right. x + 1 \left.\right) \left(\right. y + 1 \left.\right) - 9 x y}{x y} = \frac{2 - 8 x y}{x y}\)  

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

\(= \frac{2 \left(\right. x - y \left.\right)^{2}}{x y} \geq 0\)

Dấu đẳng thức xảy ra khi và chỉ khi \(x = y = \frac{1}{2}\).

x+y=1 nên \(\left(\right. 1 + \frac{1}{x} \left.\right) \left(\right. 1 + \frac{1}{y} \left.\right) - 9\)

\(= \frac{\left(\right. x + 1 \left.\right) \left(\right. y + 1 \left.\right) - 9 x y}{x y} = \frac{2 - 8 x y}{x y}\)  

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

\(= \frac{2 \left(\right. x - y \left.\right)^{2}}{x y} \geq 0\)

Dấu đẳng thức xảy ra khi và chỉ khi \(x = y = \frac{1}{2}\).

x+y=1 nên \(\left(\right. 1 + \frac{1}{x} \left.\right) \left(\right. 1 + \frac{1}{y} \left.\right) - 9\)

\(= \frac{\left(\right. x + 1 \left.\right) \left(\right. y + 1 \left.\right) - 9 x y}{x y} = \frac{2 - 8 x y}{x y}\)  

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

\(= \frac{2 \left(\right. x - y \left.\right)^{2}}{x y} \geq 0\)

Dấu đẳng thức xảy ra khi và chỉ khi \(x = y = \frac{1}{2}\).

1) \(a^{2} - a b + b^{2}\)

\(= \left(\right. a^{2} - 2 \cdot a \cdot \frac{1}{2} b + \frac{1}{4} b^{2} \left.\right) + \frac{3}{4} b^{2} = \left(\left(\right. a - \frac{1}{2} b \left.\right)\right)^{2} + \frac{3}{4} b^{2} \geq 0 \forall a , b\)

Dấu "=" xảy ra khi: \(\left{\right. a - \frac{1}{2} b = 0 \\ b = 0 \Leftrightarrow a = b = 0\) 

2) \(a^{2} - a b + b^{2} \geq \frac{1}{4} \left(\left(\right. a + b \left.\right)\right)^{2}\)

\(< = > a^{2} - a b + b^{2} \geq \frac{1}{4} \left(\right. a^{2} + 2 a b + b^{2} \left.\right) < = > a^{2} - a b + b^{2} \geq \frac{1}{4} a^{2} + \frac{1}{2} a b + \frac{1}{4} b^{2} < = > \frac{3}{4} a^{2} - \frac{3}{2} a b + \frac{3}{4} b^{2} \geq 0 < = > \frac{3}{4} \left(\right. a^{2} - 2 a b + b^{2} \left.\right) \geq 0 < = > \frac{3}{4} \left(\left(\right. a - b \left.\right)\right)^{2} \geq 0 (\text{lu} \hat{\text{o}} \text{n}\&\text{nbsp};đ \overset{ˊ}{\text{u}} \text{ng})\)

Dấu "=" xảy ra khi: \(a - b = 0 < = > a = b\)