1. **Problem statement:** Find the maximum total production (TPmax) and the labor input $L$ that achieves this maximum for the production function $$TP = -4L^2 + 64L$$. 2. **Formula and rules:** To find the maximum of a quadratic function $TP = aL^2 + bL + c$, where $a < 0$, the vertex formula gives the maximum point at $$L = -\frac{b}{2a}$$. 3. **Apply the formula:** Here, $a = -4$ and $b = 64$. So, $$L = -\frac{64}{2 \times (-4)} = -\frac{64}{-8} = 8$$. 4. **Calculate maximum production:** Substitute $L=8$ into the TP function: $$TP_{max} = -4(8)^2 + 64(8) = -4(64) + 512 = -256 + 512 = 256$$. 5. **Interpretation:** The maximum total production is 256 units, achieved when labor input $L$ is 8 units. **Final answer:** $$L = 8, \quad TP_{max} = 256$$