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 \(x^{8} - x^{7} + x^{2} - x + 1\)

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

Chú ý rằng \(x + y = 1\) 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 \(x = y = \frac{1}{2}\).

Chú ý rằng \(x + y = 1\) 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 \(x = y = \frac{1}{2}\).

Chú ý rằng \(x + y = 1\) 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 \(x = y = \frac{1}{2}\).

Chú ý rằng \(1 + 4 = 2 + 3\), ta đặt  \(t = \left(\right. x - 1 \left.\right) \left(\right. x - 4 \left.\right) = x^{2} - 5 x + 4\) thì 

\(\left(\right. x - 2 \left.\right) \left(\right. x - 3 \left.\right) = x^{2} - 5 x + 6 = t + 2\)

từ đó \(\left(\right. x - 1 \left.\right) \left(\right. x - 2 \left.\right) \left(\right. x - 3 \left.\right) \left(\right. x - 4 \left.\right) + 1\)

\(= t \left(\right. t + 2 \left.\right) + 1 = t^{2} + 2 t + 1 = \left(\right. t + 1 \left.\right)^{2} \geq 0\)

Dẳng thức chỉ xảy ra khi \(t = - 1\)

hay \(x^{2} - 5 x + 4 = - 1\)

\(x^{2} - 5 x + 5 = 0\)

\(x = \frac{5 \pm \sqrt{5}}{2}\).

Vế trái bất đẳng thức cần chứng minh là \(x^{6} \left(\right. x - 1 \left.\right)^{2} + 3 \left(\right. x^{4} - \frac{1}{2} \left.\right)^{2} + \left(\right. x - \frac{1}{2} \left.\right)^{2}\).

Từ đó suy ra đpcm.

Giả thiết đã cho tương đương với \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{a b} + \frac{1}{b c} + \frac{1}{c a} = 6\). (1)

Ta có \(\left(\right. \frac{1}{a} - 1 \left.\right)^{2} \geq 0\)

\(\frac{1}{a^{2}} + 1 \geq \frac{2}{a}\) nên 

\(\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} \geq 2 \left(\right. \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \left.\right) - 3\) (2)

Lại có \(\frac{1}{a^{2}} + \frac{1}{b^{2}} \geq \frac{2}{a b}\) nên 

\(2 \left(\right. \frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} \left.\right) \geq 2 \left(\right. \frac{1}{a b} + \frac{1}{b c} + \frac{1}{c a} \left.\right)\) (3)

Cộng (2) và (3) theo vế  và sử dụng (1)  ta có    

\(3 \left(\right. \frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} \left.\right) \geq 2 \left(\right. \frac{1}{a b} + \frac{1}{b c} + \frac{1}{c a} + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \left.\right) - 3 = 2.6 - 3 = 9\) 

Suy ra \(\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} \geq 3\).

Ta có \(x^{2} + y^{2} + x y - 3 x - 3 y + 3\)

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

\(= \left(\right. x - 1 \left.\right)^{2} + \left(\right. y - 1 \left.\right)^{2} + \left(\right. x - 1 \left.\right) \left(\right. y - 1 \left.\right) \geq 0\)                       

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

Ta có \(x^{2} + y^{2} + x y - 3 x - 3 y + 3\)

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

\(= \left(\right. x - 1 \left.\right)^{2} + \left(\right. y - 1 \left.\right)^{2} + \left(\right. x - 1 \left.\right) \left(\right. y - 1 \left.\right) \geq 0\)                       

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

Ta có \(\sqrt{a^{2} - a b + b^{2}} = \sqrt{\frac{1}{4} \left(\right. a + b \left.\right)^{2} + \frac{3}{4} \left(\right. a - b \left.\right)^{2}} \&\text{nbsp}; \geq \frac{1}{2} \left(\right. a + b \left.\right)\).

Đẳng thức xảy ra khi và chỉ khi \(a = b\).

Trương tự \(\sqrt{b^{2} - b c + c^{2}} \geq \frac{1}{2} \left(\right. b + c \left.\right)\) và \(\sqrt{c^{2} - c a + c a} \geq \frac{1}{2} \left(\right. c + a \left.\right)\).

Từ đó \(\sqrt{a^{2} - a b + b^{2}} + \sqrt{b^{2} - b c + c^{2}} + \sqrt{c^{2} - c a + a^{2}} \geq \frac{1}{2} \left(\right. a + b + b + c + c + a \left.\right)\)

\(= \left(\right. a + b + c \left.\right) = 3\)                                                                           

Vậy \(\sqrt{a^{2} - a b + b^{2}} + \sqrt{b^{2} - b c + c^{2}} + \sqrt{c^{2} - c a + a^{2}} \geq 3\).

Đẳng thức xảy ra khi và chỉ khi \(a = b = c = \frac{a + b + c}{3} = 1\).