30km/1h

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 \(x^{8} - x^{7} + x^{2} - x + 1\)

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

0a2−ab+b2=41​(4a2−4ab+4b2)=41​(2a−b)2+43​b2≥0.

Đẳng thức xảy ra khi và chỉ khi

hay \(a = b = 0\).

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

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

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

ta có \(y \left(\right. y - z \left.\right) \left(\right. y - x \left.\right) + z \left(\right. z - x \left.\right) \left(\right. z - y \left.\right) = \left(\right. z - y \left.\right) \left[\right. z \left(\right. z - x \left.\right) - y \left(\right. y - x \left.\right) \left]\right.\) (3)

Mà \(z \geq y \geq x \geq 0\) nên \(z \geq y \geq 0\) và \(z - x \geq y - x \geq 0\), từ đó  

\(z \left(\right. z - x \left.\right) \geq y \left(\right. y - x \left.\right)\) nên \(z \left(\right. z - x \left.\right) - y \left(\right. y - x \left.\right) \geq 0\) (4)

Từ (2) và (4) suy ra  \(\left(\right. z - y \left.\right) \left[\right. z \left(\right. z - x \left.\right) - y \left(\right. y - x \left.\right) \left]\right. \geq 0\), kết hợp với (3) suy ra 

\(y\left(\right.y-z\left.\right)\left(\right.y-x\left.\right)+z\left(\right.z-x\left.\right)\left(\right.z=-y\left.\right)\geq0\)