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 — Here's how to determine the nature of the stationary points for y = 8x^3 + 15x^2 - 18x + 12. Step 1: Find the first derivative, (dy)/(dx). (dy)/(dx) = (d)/(dx)(8x^3 + 15x^2 - 18x + 12) = 24x^2 + 30x - 18 Step 2: Find the x-coordinates of the stationary points by setting the first derivative to zero. 24x^2 + 30x - 18 = 0 Divide the entire equation by 6 to simplify: 4x^2 + 5x - 3 = 0 Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a with a=4, b=5, c=-3: x = -5 ± sqrt(5^2 - 4(4)(-3))2(4) x = -5 ± sqrt(25 + 48)8 x = -5 ± sqrt(73)8 The stationary points occur at x_1 = -5 + sqrt(73)8 and x_2 = -5 - sqrt(73)8. Step 3: Find the second derivative, (d^2y)/(dx^2). (d^2y)/(dx^2) = (d)/(dx)(24x^2 + 30x - 18) = 48x + 30 Step 4: Determine the nature of the stationary points using the second derivative test. For x_1 = -5 + sqrt(73)8: Substitute x_1 into the second derivative: (d^2y)/(dx^2) |_x=-5 + sqrt(73)8 = 48(-5 + sqrt(73)8) + 30 = 6(-5 + sqrt(73)) + 30 = -30 + 6sqrt(73) + 30 = 6sqrt(73) Since 6sqrt(73) > 0, the stationary point at x = -5 + sqrt(73)8 is a local minimum. For x_2 = -5 - sqrt(73)8: Substitute x_2 into the second derivative: (d^2y)/(dx^2) |_x=-5 - sqrt(73)8 = 48(-5 - sqrt(73)8) + 30 = 6(-5 - sqrt(73)) + 30 = -30 - 6sqrt(73) + 30 = -6sqrt(73) Since -6sqrt(73) < 0, the stationary point at x = -5 - sqrt(73)8 is a local maximum. Drop the next question!