In the question, two quantities i.e. Quantity I and Quantity II are given. Solve the given quantities to establish the correct relation between them and choose the correct option.

Quantity I:  a²c + ab² - ac² - ba² + bc² – cb + abc

Quantity II:  (ab + bc) (c² - b² - a²)

Solution

ATQ, a²c + ab²- ac²- ba² + bc²- cb² + abc (a - b)(b - c)(c - a) + abc Minimum value, (2 - 5)(5 - 8)(8 - 2) + 2 × 5 × 8 = (- 3)(- 3) × 6 + 80 = 54 + 80 = 134 Maximum value, (4 - 7)(7 - 10)(10 - 4) + 4 × 7 × 10 = (- 3)(- 3) × 6 + 280 = 54 + 280 = 334 Quantity II (ab + bc)(c²- b²- c²) Minimum value, (2 × 5 + 5 × 8) × (82 - 52 - 22) = (10 + 40) × (64 - 25 - 4) = 50 × 35 = 1750 Maximum value, (4 × 7 + 7 × 10) × (102 - 72 - 42) = (28 + 70) × (100 - 49 - 16) = 98 × 35 = 3430 Hence, Quantity I < Quantity II .