Please explain why is it the short run production function, how do we know it? thanks

* Consider a production function Q = F (K, L) = 5L + L2K1/2 L3. lf capital is xed at K0 = 4, what value of L maximises the average product of labour? At what value of L is the average product of labour equal to the marginal product of labour? ANSWER: The short run production function is given by Q = 5L + 2L2 L3. The average product of labour is? = 5 + 2L L2. A necessary condition for a maximum of the average product of labour is that its rst derivative with respect to L is 0. The rst derivative is d(Q/L)/dL = 2 2L. Setting this to 0 and checking the second- order condition, we can see that the average product of labour is maximised where L = 1. The marginal product of labour is is dQ /dL = 5 + 4L 3L2. Setting this equal to the average product of labour, we can see that the two expressions are equal where L = 1. That is, they are equal at the maximum of the average product of labour. This is intuitive because APL is increasing where MP1L > APL, and it is decreasing where MPL