answer:To find the inflection points of the function f(x) = x^3 - 6x^2 + 9x, we first need to find the second derivative of the function. First, let's find the first derivative, f'(x): f'(x) = d/dx (x^3 - 6x^2 + 9x) = 3x^2 - 12x + 9 Now, let's find the second derivative, f''(x): f''(x) = d/dx (3x^2 - 12x + 9) = 6x - 12 To find the inflection points, we need to find the values of x where f''(x) = 0: 6x - 12 = 0 6x = 12 x = 2 Now we need to check if this value of x is within the given interval [0, 5]. Since 2 is within the interval [0, 5], we have one inflection point at x = 2. To find the corresponding y-value, we plug x = 2 back into the original function f(x): f(2) = (2)^3 - 6(2)^2 + 9(2) = 8 - 24 + 18 = 2 So, the inflection point within the interval [0, 5] is at (2, 2).

answer:To find the inflection point(s) of the profit function, we first need to find the second derivative of the function and then set it equal to zero to find the critical points. The first derivative of the function f(x) is: f'(x) = 3x^2 - 12x + 9 The second derivative of the function f(x) is: f''(x) = 6x - 12 Now, we set the second derivative equal to zero to find the critical points: 6x - 12 = 0 Solve for x: x = 2 Now that we have the critical point, we need to check if it's an inflection point by analyzing the concavity of the function on either side of the critical point. If the concavity changes, then it is an inflection point. Choose a test point to the left of x = 2, say x = 1: f''(1) = 6(1) - 12 = -6 (which is negative, so the function is concave down) Choose a test point to the right of x = 2, say x = 3: f''(3) = 6(3) - 12 = 6 (which is positive, so the function is concave up) Since the concavity changes from concave down to concave up at x = 2, we have an inflection point at x = 2. Now, we find the corresponding profit value at this inflection point: f(2) = (2)^3 - 6(2)^2 + 9(2) - 2 = 8 - 24 + 18 - 2 = 0 So, the inflection point is (2, 0). In the context of the problem, this inflection point indicates that at a production level of 2,000 units (x = 2), the profit earned by the company is 0 (f(2) = 0). Additionally, the inflection point signifies a change in the rate of profit growth. Before producing 2,000 units, the profit growth was slowing down (concave down), but after producing 2,000 units, the profit growth starts to accelerate (concave up). This suggests that the company should produce more than 2,000 units to maximize their profit growth.

answer:To find the inflection points of the function f(x) = x^3 - 6x^2 + 9x + 2, we need to find the second derivative of the function and then solve for x when the second derivative is equal to 0. First, let's find the first derivative, f'(x): f'(x) = d/dx (x^3 - 6x^2 + 9x + 2) = 3x^2 - 12x + 9 Now, let's find the second derivative, f''(x): f''(x) = d/dx (3x^2 - 12x + 9) = 6x - 12 To find the inflection points, we need to solve for x when f''(x) = 0: 6x - 12 = 0 6x = 12 x = 2 So, there is an inflection point at x = 2. To find the corresponding y-coordinate, plug x = 2 back into the original function f(x): f(2) = (2)^3 - 6(2)^2 + 9(2) + 2 = 8 - 24 + 18 + 2 = 4 Thus, the inflection point of the function f(x) = x^3 - 6x^2 + 9x + 2 is (2, 4).

answer:To find the inflection points of the function f(x) = 3x^3 - 10x^2 + 5, we first need to find the second derivative of the function and then set it to zero to find the critical points. 1. Find the first derivative, f'(x): f'(x) = d/dx(3x^3 - 10x^2 + 5) = 9x^2 - 20x 2. Find the second derivative, f''(x): f''(x) = d/dx(9x^2 - 20x) = 18x - 20 3. Set the second derivative to zero and solve for x: 0 = 18x - 20 20 = 18x x = 20/18 = 10/9 4. Plug the x-value back into the original function to find the corresponding y-value: f(10/9) = 3(10/9)^3 - 10(10/9)^2 + 5 f(10/9) = 3(1000/729) - 10(100/81) + 5 f(10/9) = 3000/729 - 1000/81 + 5 f(10/9) = 4050/729 - 1000/81 f(10/9) = 1350/729 f(10/9) = 50/27 So, the inflection point is (10/9, 50/27).