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
Here's how to find the stationary points for 2(a)(i): Step 1: Find the first derivative of the function. Given y = 2x^3 - 9x^2 + 12x. (dy)/(dx) = (d)/(dx)(2x^3 - 9x^2 + 12x) (dy)/(dx) = 6x^2 - 18x + 12 Step 2: Set the first derivative to zero to find the x-coordinates of the stationary points. 6x^2 - 18x + 12 = 0 Divide the equation by 6: x^2 - 3x + 2 = 0 Step 3: Solve the quadratic equation for x. Factor the quadratic equation: (x-1)(x-2) = 0 This gives two possible values for x: x-1 = 0 x = 1 x-2 = 0 x = 2 Step 4: Substitute these x-values back into the original function y = 2x^3 - 9x^2 + 12x to find the corresponding y-coordinates. For x=1: y = 2(1)^3 - 9(1)^2 + 12(1) y = 2 - 9 + 12 y = 5 So, the first stationary point is (1, 5). For x=2: y = 2(2)^3 - 9(2)^2 + 12(2) y = 2(8) - 9(4) + 24 y = 16 - 36 + 24 y = 4 So, the second stationary point is (2, 4). The stationary points are (1, 5) and (2, 4). That's 2 down. 3 left today — send the next one.