Total cost and market demand of a monopoly firm: C=(1/3)x³−10x²+9x+1000; x=(12−p)/9. At which output is…

From the demand equation x = (12 - p)/9, the inverse demand function is p = 12 - 9x. 1. Total Revenue: TR = p * x = (12 - 9x)x = 12x - 9x² 2. Marginal Revenue: MR = d(TR)/dx = 12 - 18x 3. Marginal Cost: MC = dC/dx = d/dx [(1/3)x³ - 10x² + 9x + 1000] = x² - 20x + 9 4. First-order condition for profit maximisation (MR = MC): x² - 20x + 9 = 12 - 18x => x² - 2x - 3 = 0 => (x - 3)(x + 1) = 0. Since output x must be positive, x = 3. 5. Second-order condition: d²π/dx² = d(MR - MC)/dx = -18 - (2x - 20) = 2 - 2x. At x = 3, d²π/dx² = 2 - 6 = -4 < 0, confirming that profit is maximised at x = 3 .