I need to explain fully how and why the firms the product and cost curves are functionally related thank you
The formula for production function is Q = f(K, L). Quantity is a function of the inputs used to produce it: capital and labor. In the other hand, total costs of production equals to the cost of all inputs used, so: TC = Pk*K + PL*L, where Pk - price of capital (or rent) and PL - price of labor (or wage rate). The long run total cost function C = g(Q) depends on the prices of inputs, it can be a good proxy for the opportunity cost of delivering Q, at least where we measure costs in units of present value.