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
Step 1: Find the first derivative of f(x). Given f(x) = 2x^3 - 5x^2 - 4x + 3. To find the turning points, we set the first derivative equal to zero. f'(x) = (d)/(dx)(2x^3 - 5x^2 - 4x + 3) = 6x^2 - 10x - 4 Step 2: Set f'(x) = 0 and solve for x. 6x^2 - 10x - 4 = 0 Divide by 2 to simplify: 3x^2 - 5x - 2 = 0 Factor the quadratic equation: (3x+1)(x-2) = 0 This gives two possible x-coordinates for the turning points: 3x+1 = 0 x = -(1)/(3) x-2 = 0 x = 2 Step 3: Find the second derivative f''(x) to classify the turning points. f''(x) = (d)/(dx)(6x^2 - 10x - 4) = 12x - 10 Evaluate f''(x) at each x-value: For x = -(1)/(3): f''(-(1)/(3)) = 12(-(1)/(3)) - 10 = -4 - 10 = -14 Since f''(-(1)/(3)) < 0, this corresponds to a local maximum (point M). For x = 2: f''(2) = 12(2) - 10 = 24 - 10 = 14 Since f''(2) > 0, this corresponds to a local minimum (point N). Step 4: Determine the y-coordinate of the minimum turning point N. Substitute x=2 into the original function f(x): f(2) = 2(2)^3 - 5(2)^2 - 4(2) + 3 f(2) = 2(8) - 5(4) - 8 + 3 f(2) = 16 - 20 - 8 + 3 f(2) = -4 - 8 + 3 f(2) = -12 + 3 f(2) = -9 The coordinates of the minimum turning point N are (2; -9). The coordinates of N are (2; -9). That's 2 down. 3 left today — send the next one.