1. **State the problem:** We have a cost function for producing $x$ items: $$C(x) = 2x^2 + 3x + 98$$ We need to find: - Average cost function - Marginal cost function - Marginal average cost function - Marginal cost and marginal average cost at $x=5$ - The value of $x$ where marginal cost equals average cost 2. **Formulas and rules:** - Average cost $AC(x) = \frac{C(x)}{x}$ for $x > 0$ - Marginal cost $MC(x) = C'(x)$, the derivative of the cost function - Marginal average cost $MAC(x) = AC'(x)$, the derivative of the average cost 3. **Find average cost:** $$AC(x) = \frac{2x^2 + 3x + 98}{x} = 2x + 3 + \frac{98}{x}$$ 4. **Find marginal cost:** $$MC(x) = \frac{d}{dx}(2x^2 + 3x + 98) = 4x + 3$$ 5. **Find marginal average cost:** $$AC(x) = 2x + 3 + 98x^{-1}$$ $$MAC(x) = \frac{d}{dx}(2x + 3 + 98x^{-1}) = 2 - 98x^{-2} = 2 - \frac{98}{x^2}$$ 6. **Evaluate marginal cost and marginal average cost at $x=5$:** $$MC(5) = 4(5) + 3 = 20 + 3 = 23$$ $$MAC(5) = 2 - \frac{98}{5^2} = 2 - \frac{98}{25} = 2 - 3.92 = -1.92$$ 7. **Find $x$ such that marginal cost equals average cost:** Set $MC(x) = AC(x)$: $$4x + 3 = 2x + 3 + \frac{98}{x}$$ Simplify: $$4x + 3 - 2x - 3 = \frac{98}{x}$$ $$2x = \frac{98}{x}$$ Multiply both sides by $x$: $$2x^2 = 98$$ $$x^2 = 49$$ $$x = 7$$ (since $x > 0$) **Final answers:** - Average cost: $$AC(x) = 2x + 3 + \frac{98}{x}$$ - Marginal cost: $$MC(x) = 4x + 3$$ - Marginal average cost: $$MAC(x) = 2 - \frac{98}{x^2}$$ - At $x=5$: $$MC(5) = 23$$, $$MAC(5) = -1.92$$ - $x$ where $MC = AC$: $$x = 7$$