This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — The given function is the derivative of f(x): f'(x) = x^2 + 2x - 3 a) Determine coordinates of the turning point. Step 1: Find the x-coordinates of the turning points of f(x) by setting f'(x) = 0. x^2 + 2x - 3 = 0 Factor the quadratic equation: (x+3)(x-1) = 0 This gives two x-coordinates for the turning points: x = -3 or x = 1 Step 2: Find the function f(x) by integrating f'(x). f(x) = (x^2 + 2x - 3)\,dx f(x) = x^2+12+1 + 2x^1+11+1 - 3x + C f(x) = (x^3)/(3) + x^2 - 3x + C Step 3: Substitute the x-coordinates into f(x) to find the corresponding y-coordinates. For x = -3: f(-3) = ((-3)^3)/(3) + (-3)^2 - 3(-3) + C f(-3) = (-27)/(3) + 9 + 9 + C f(-3) = -9 + 9 + 9 + C f(-3) = 9 + C The first turning point is (-3, 9+C). For x = 1: f(1) = ((1)^3)/(3) + (1)^2 - 3(1) + C f(1) = (1)/(3) + 1 - 3 + C f(1) = (1)/(3) - 2 + C f(1) = -(5)/(3) + C The second turning point is (1, -(5)/(3)+C). The coordinates of the turning points are (-3, 9+C) and (1, -(5)/(3)+C). b) Determine the interval over which f is increasing and decreasing. Step 1: Recall that f(x) is increasing when f'(x) > 0 and decreasing when f'(x) < 0. We found the roots of f'(x) = 0 to be x = -3 and x = 1. Since f'(x) = x^2 + 2x - 3 is a parabola opening upwards (the coefficient of x^2 is positive), f'(x) is positive outside its roots and negative between its roots. Step 2: Determine the intervals. f(x) is increasing when f'(x) > 0: x^2 + 2x - 3 > 0 (x+3)(x-1) > 0 This occurs when x < -3 or x > 1. $f(x