This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.
ScanSolve AI Answer
Step 1: Find the first derivative of the function. The given function is y = (1)/(3)x^3 - x + 7. To find the turning points, we first calculate the first derivative (dy)/(dx). (dy)/(dx) = (d)/(dx)((1)/(3)x^3 - x + 7) (dy)/(dx) = (1)/(3)(3x^2) - 1 + 0 (dy)/(dx) = x^2 - 1 Step 2: Set the first derivative to zero to find the critical points. x^2 - 1 = 0 x^2 = 1 x = ±sqrt(1) So, x = 1 or x = -1. These are the x-coordinates of the turning points. Step 3: Find the y-coordinates of the turning points. Substitute the x-values back into the original function y = (1)/(3)x^3 - x + 7. For x = 1: y = (1)/(3)(1)^3 - (1) + 7 y = (1)/(3) - 1 + 7 y = (1)/(3) + 6 y = (1)/(3) + (18)/(3) y = (19)/(3) The first turning point is (1, (19)/(3)). For x = -1: y = (1)/(3)(-1)^3 - (-1) + 7 y = (1)/(3)(-1) + 1 + 7 y = -(1)/(3) + 8 y = -(1)/(3) + (24)/(3) y = (23)/(3) The second turning point is (-1, (23)/(3)). Step 4: Find the second derivative to classify the turning points. The first derivative is (dy)/(dx) = x^2 - 1. (d^2y)/(dx^2) = (d)/(dx)(x^2 - 1) (d^2y)/(dx^2) = 2x Step 5: Classify the turning points using the second derivative test. For x = 1: (d^2y)/(dx^2) = 2(1) = 2 Since (d^2y)/(dx^2) > 0, the turning point (1, (19)/(3)) is a local minimum. For x = -1: (d^2y)/(dx^2) = 2(-1) = -2 Since (d^2y)/(dx^2) < 0, the turning point (-1, (23)/(3)) is a local maximum. The turning points are: Local minimum: (1, (19)/(3)) Local maximum: (-1, (23)/(3)) That's 2 down. 3 left today — send the next one.