Question

Доведіть, що вираз 3x²(4-4x)-4x(3x+3x²-x³) набуває невід'ємних значень при всiх значеннях змiнної х.

Answer 1

Proof that the expression 3x²(4-4x)-4x(3x+3x²-x³) takes non-negative values for all values of variable x.

To prove that the expression 3x²(4-4x)-4x(3x+3x²-x³) takes non-negative values for all values of the variable x, we need to show that the expression is greater than or equal to zero for any value of x.

Let's simplify the expression step by step:

1. Distribute the terms inside the parentheses: 3x²(4-4x) - 4x(3x+3x²-x³) = 12x² - 12x³ - 12x³ - 12x⁴ - 4x³ + 4x⁴

2. Combine like terms: 12x² - 12x³ - 12x³ - 12x⁴ - 4x³ + 4x⁴ = 12x² - 16x³ - 8x⁴

3. Factor out common terms: 12x² - 16x³ - 8x⁴ = 4x²(3 - 4x - 2x²)

Now, we need to prove that the expression 4x²(3 - 4x - 2x²) is non-negative for all values of x.

To do this, we can analyze the factors separately:

1. 4x² is always non-negative because it is the square of a real number.

2. (3 - 4x - 2x²) is a quadratic expression. To determine its sign, we can find the roots of the equation (3 - 4x - 2x²) = 0.

Using the quadratic formula, we can find the roots of the equation: x = (-(-4) ± √((-4)² - 4(2)(3))) / (2(2)) x = (4 ± √(16 - 24)) / 4 x = (4 ± √(-8)) / 4

Since the discriminant (√(-8)) is negative, the quadratic expression (3 - 4x - 2x²) does not have real roots. This means that the expression is either always positive or always negative.

To determine the sign of (3 - 4x - 2x²), we can analyze its leading coefficient (which is negative) and the sign of the coefficient of the x term (which is negative as well).

Based on these factors, we can conclude that (3 - 4x - 2x²) is always negative.

Therefore, the expression 4x²(3 - 4x - 2x²) is non-negative for all values of x, since it is the product of a non-negative factor (4x²) and a negative factor (3 - 4x - 2x²).

In conclusion, we have proven that the expression 3x²(4-4x)-4x(3x+3x²-x³) takes non-negative values for all values of the variable x.