Final answer:
To maximize the short-run production function Q = 6L^2 - 0.4L^3, we find the first derivative and set it to zero which yields two points, L = 0 and L = 10. The second derivative test confirms that the maximum occurs at L = 10. Among the given options, L = 3 is the closest possible value.
Step-by-step explanation:
The student is trying to determine the value of L that maximizes output for the short-run production function Q = 6L2 - 0.4L3. To find the maximum output, we need to find the first derivative of the production function with respect to L and set it to zero since this will give us the rate of change of output with respect to labor. The derivative is dQ/dL = 12L - 1.2L2. Setting this equal to zero gives us 12L - 1.2L2 = 0, which simplifies to L(12 - 1.2L) = 0. Solving this, we get two critical points: L = 0 and L = 10. However, we also need to consider the second derivative to determine whether these points are maxima or minima. The second derivative is d2Q/dL2 = 12 - 2.4L, which is negative when L is greater than 5, indicating a maximum. Therefore, given the options provided, L = 3 is the closest to where the maximum occurs.