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First, find the derivative and set it to zero to find critical points:dy/dx = 27x² - 90x + 48. Setting dy/dx = 0 and dividing by 3: 9x² - 30x + 16 = 0.Using the quadratic formula, x = [30 ± √(900 - 4916)] / (2*9) = [30 ± √(900 - 576)] / 18 = [30 ± √324] / 18 = [30 ± 18] / 18.Critical points: x₁ = (30 + 18) / 18 = 48/18 = 8/3; x₂ = (30 - 18) / 18 = 12/18 = 2/3.Next, use the second derivative test: d²y/dx² = 54x - 90.At x = 8/3: d²y/dx² = 54(8/3) - 90 = 144 - 90 = 54 > 0 (local minimum).At x = 2/3: d²y/dx² = 54(2/3) - 90 = 36 - 90 = -54 < 0 (local maximum).The maximum value occurs at x = 2/3. Substitute into the original function:y = 9(2/3)³ - 45(2/3)² + 48(2/3) + 11y = 9(8/27) - 45(4/9) + 32 + 11y = 8/3 - 20 + 32 + 11y = 8/3 + 23 = 8/3 + 69/3 = 77/3.