answer:This problem tests knowledge of derivative rules, determination of a function's zeros, the definition of a "related function", and properties of quadratic functions. It reflects a mathematical thinking of transformation and is of medium difficulty. First, differentiate f(x). According to the problem, the function h(x) = f'(x) - g(x) = x(2x - 5x + 4 - m) has two distinct zeros in [0, 3]. Therefore, we have the following conditions: begin{cases} h(0) geq 0 h(3) geq 0 h(frac{5}{2}) < 0 end{cases} Solving these inequalities will give us the range of m. Analysis: 1. Differentiate f(x): f'(x) = frac{2x}{3} - 3x + 4 2. Set up the function h(x) = f'(x) - g(x) = x(2x - 5x + 4 - m): h(x) = x(2 - 3 - m) + 4x 3. Solve for the conditions: begin{cases} h(0) = 4 - m geq 0 h(3) = -2 - m geq 0 h(frac{5}{2}) = frac{25}{4} - frac{25}{2} + 4 - m < 0 end{cases} 4. Solve the system of inequalities to find the range of m: -frac{9}{2} < m leq -2 So, the answer is (A): boxed{left(-frac{9}{4}, -2right)}.

answer:We start with the given quadratic equation in terms of (x) and (y): [ x^2 + xy + 2y^2 = 29. ] 1. **Rewrite the equation as a quadratic in (x):** [ x^2 + xy + (2y^2 - 29) = 0. ] 2. **Determine the condition for integer solutions:** For (x) to have integer solutions, the discriminant (Delta) of the quadratic equation must be a perfect square. For a quadratic equation of the form (ax^2 + bx + c = 0), the discriminant (Delta) is given by: [ Delta = b^2 - 4ac. ] In our case: [ a = 1, ] [ b = y, ] [ c = 2y^2 - 29. ] Therefore, the discriminant is: [ Delta = y^2 - 4 cdot 1 cdot (2y^2 - 29). ] 3. **Simplify the discriminant:** [ Delta = y^2 - 4(2y^2 - 29) Delta = y^2 - 8y^2 + 116 Delta = -7y^2 + 116. ] 4. **Ensure the discriminant is non-negative:** For (Delta) to be a perfect square, it must be non-negative: [ -7y^2 + 116 geq 0 116 geq 7y^2 y^2 leq frac{116}{7} y^2 leq 16.57 y^2 leq 16. ] 5. **Determine the integer values of (y^2):** Since (y^2) must be a perfect square, the possible integer values of (y^2) within this range are: [ y^2 = 16, 9, 4, 1, 0. ] 6. **Check which values of (y^2) yield a perfect square discriminant:** We will now verify these values individually to see if they result in (Delta) being a perfect square. - For (y^2 = 16): [ Delta = -7 cdot 16 + 116 = -112 + 116 = 4. ] Since (Delta = 4), which is a perfect square, this value of (y) works. - For (y^2 = 9): [ Delta = -7 cdot 9 + 116 = -63 + 116 = 53. ] Since (Delta = 53), which is not a perfect square, this value of (y) does not work. - For (y^2 = 4): [ Delta = -7 cdot 4 + 116 = -28 + 116 = 88. ] Since (Delta = 88), which is not a perfect square, this value of (y) does not work. - For (y^2 = 1): [ Delta = -7 cdot 1 + 116 = -7 + 116 = 109. ] Since (Delta = 109), which is not a perfect square, this value of (y) does not work. - For (y^2 = 0): [ Delta = -7 cdot 0 + 116 = 0 + 116 = 116. ] Since (Delta = 116), which is not a perfect square, this value of (y) does not work. The only valid solution (Delta = 4) corresponds to (y^2 = 16). Thus, (y = 4) or (y = -4). 7. **Find the corresponding (x) values for (y = 4) and (y = -4):** For (y = 4), the equation becomes: [ x^2 + 4x + 2 cdot 4^2 - 29 = 0 x^2 + 4x + 3 = 0. ] Solving for (x): [ x = frac{-4 pm sqrt{16 - 12}}{2} x = frac{-4 pm 2}{2} x = -1 text{ or } x = -3. ] For (y = -4), the equation becomes: [ x^2 - 4x + 2 cdot (-4)^2 - 29 = 0 x^2 - 4x + 3 = 0. ] Solving for (x): [ x = frac{4 pm sqrt{16 - 12}}{2} x = frac{4 pm 2}{2} x = 1 text{ or } x = 3. ] Therefore, the integer solutions ((x, y)) are: [ (x_1, y_1) = (-1, 4), (x_2, y_2) = (-3, 4), (x_3, y_3) = (1, -4), (x_4, y_4) = (3, -4). ] Thus, there are 4 integer solutions. Conclusion: [boxed{text{C}}]

answer:To find the sum of the numerical coefficients in the expansion of (a+b)^8, we apply the same method as was used for a sixth-degree binomial. Substitute a=1 and b=1 in (a+b)^8. 1. Substitute a = 1 and b = 1 into (a+b)^8: [ (1+1)^8 ] 2. Calculate the simplified expression: [ 2^8 = 256 ] Therefore, the sum of the numerical coefficients in the expansion of (a+b)^8 is 256. 256 The correct answer is boxed{textbf{(B)} 256}.

answer:Let's solve this step by step using algebra. 1. We are given that 45% of z is 72% of y: 0.45z = 0.72y 2. y is 75% of x: y = 0.75x 3. w is 60% of z^2: w = 0.60z^2 4. z is 30% of w^(1/3): z = 0.30w^(1/3) 5. v is 80% of x^0.5: v = 0.80x^0.5 We need to find what percent of the sum of x and v is z. Let's express everything in terms of x. From (2), we can express y in terms of x: y = 0.75x Substitute y in (1) to get z in terms of x: 0.45z = 0.72 * 0.75x 0.45z = 0.54x z = (0.54/0.45)x z = 1.2x Now, let's find w in terms of z using (3): w = 0.60z^2 w = 0.60(1.2x)^2 w = 0.60 * 1.44x^2 w = 0.864x^2 Using (4), we can express z in terms of w: z = 0.30w^(1/3) z = 0.30(0.864x^2)^(1/3) z = 0.30(0.936x^(2/3)) z = 0.2808x^(2/3) We have two expressions for z in terms of x, which should be equal: 1.2x = 0.2808x^(2/3) To solve for x, we can equate the two expressions and solve for x: 1.2x = 0.2808x^(2/3) x^(1/3) = 0.2808 / 1.2 x^(1/3) = 0.234 x = 0.234^3 x ≈ 0.0128 Now, let's find v in terms of x using (5): v = 0.80x^0.5 v = 0.80 * 0.0128^0.5 v ≈ 0.80 * 0.1131 v ≈ 0.0905 Now we have x and v, we can find the sum of x and v: Sum = x + v Sum ≈ 0.0128 + 0.0905 Sum ≈ 0.1033 Finally, we need to find what percent of the sum is z. We already have z in terms of x: z = 1.2x z ≈ 1.2 * 0.0128 z ≈ 0.01536 Now, we find the percentage: Percentage = (z / Sum) * 100 Percentage ≈ (0.01536 / 0.1033) * 100 Percentage ≈ 14.86% Therefore, z is approximately boxed{14.86%} of the sum of x and v.