1. **State the problem:** We are given the cost function $$C(q) = 500 + 3q + q^2$$ and asked to find the second derivative $$C''(5)$$. 2. **Recall the formula:** The first derivative $$C'(q)$$ represents the rate of change of cost with respect to quantity $$q$$. The second derivative $$C''(q)$$ represents the rate of change of the first derivative, or the acceleration of cost change. 3. **Find the first derivative:** $$C'(q) = \frac{d}{dq}(500 + 3q + q^2) = 0 + 3 + 2q = 3 + 2q$$ 4. **Find the second derivative:** $$C''(q) = \frac{d}{dq}(3 + 2q) = 0 + 2 = 2$$ 5. **Evaluate at $$q=5$$:** $$C''(5) = 2$$ **Final answer:** $$C''(5) = 2$$