Final answer:
To find the maximum of the function f(x) = 4x - 1.8x² + 1.2x³ - 0.3x⁴ using the golden-section search method, we start with an initial interval and iteratively narrow it down until a termination criterion is met. The maximum point is the upper bound of the final interval.
Step-by-step explanation:
The given function is f(x) = 4x - 1.8x² + 1.2x³ - 0.3x⁴. To find the maximum of this function using the golden-section search method, we start with an interval [x₁, xu] = [-2, 4]. The termination criterion is given as εₛ = 1%.
Step 1: Calculate the values of the function at the endpoints of the interval. f(x₁) = 4(-2) - 1.8(-2)² + 1.2(-2)³ - 0.3(-2)⁴ = -16 and f(xu) = 4(4) - 1.8(4)² + 1.2(4)³ - 0.3(4)⁴ = 68.48.
Step 2: Calculate the values of the function at two internal points. Let's choose one internal point x₂ as x₂ = xu - φ(xu - x₁) = 4 - 0.618(4 - (-2)) ≈ 0.472. Evaluating f(x₂) gives f(x₂) = 4(0.472) - 1.8(0.472)² + 1.2(0.472)³ - 0.3(0.472)⁴ ≈ 1.702.
Step 3: Update the interval based on the values of the function. Since f(x₂) is greater than both f(x₁) and f(xu), we update the interval to [x₁, xu] = [x₁, x₂]. The new interval becomes [-2, 0.472]. Repeat steps 1-3 until the width of the interval is less than εₛ (1% of the original interval).
Continue this process iteratively, narrowing down the interval until the termination criterion is met. The maximum point for the function is the upper bound of the final interval.