Math: Cost Functions Consider a firm in a competitive market. Given that its total cost function c(q) is increasing and smooth (first and second derivatives are continuous) with c(0) = 0, and MC function is first decreasing and then increasing in q. (1) Show that the point where MC and AC cross with each other can only lie on the upward-sloping part of the MC curve. (Hint: Mean value theorem says, for any differentiable function f (@) on a closed interval [a, b], there exists a point c E (a, b) such that f'(c) = [f(b) - f(a)]/(b - a). Use this theorem twice). (2) Show that this point is the minimum point of AC. (Hint: To show the first order derivative is zero and the second order derivative is positive). (3) Show that this point is the profit-maximizing quantity in the long run