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 the function. Given the function y = 2x^3 + 5. Differentiate y with respect to x: (dy)/(dx) = (d)/(dx)(2x^3 + 5) = 6x^2 Step 2: Find the x-coordinates of the stationary points by setting the first derivative to zero. 6x^2 = 0 x^2 = 0 x = 0 There is one stationary point at x=0. Step 3: Find the y-coordinate of the stationary point by substituting x=0 into the original equation. y = 2(0)^3 + 5 = 0 + 5 = 5 The stationary point is (0, 5). Step 4: Find the second derivative to determine the nature of the stationary point. Differentiate (dy)/(dx) = 6x^2 with respect to x: (d^2y)/(dx^2) = (d)/(dx)(6x^2) = 12x Step 5: Evaluate the second derivative at the stationary point x=0. (d^2y)/(dx^2) |_x=0 = 12(0) = 0 Since the second derivative is 0, we need to check the sign of the first derivative around x=0 to confirm the nature of the point. For x < 0 (e.g., x = -1), (dy)/(dx) = 6(-1)^2 = 6 > 0. For x > 0 (e.g., x = 1), (dy)/(dx) = 6(1)^2 = 6 > 0. Since the sign of (dy)/(dx) does not change around x=0 (it remains positive), the point (0, 5) is a point of inflection. The stationary point is (0, 5) which is a point of inflection. Send me the next one 📸