Production Level and a Cost Parameter
A workshop models daily profit in hundreds of yuan by f(x)=x^3-3ax+1, where x in [0,2] is the production level in hundreds of units and a is a nonnegative supplier-cost parameter. The current contract sets a=1. The manager says: 'Because the cubic term dominates, pushing production to x=2 always maximizes profit, and raising a only shifts the curve down without changing where the maximum occurs.' An engineer replies that for some contracts the best choice is x=0. The board wants a complete range analysis before renewing the contract, including whether the manager's claims hold at a=1 and in general.
1. Find f'(x) and all stationary points in (0,2) for a ≥ 0. Classify them and state how the number changes with a.
Read the worked answer
f'(x)=3x^2−3a=3(x^2−a). On the open interval (0,2): if a=0, f'(x)=3x^2>0, so no stationary point; if 0<a<4, the only stationary point is x=√a; if a≥4, there is no stationary point in (0,2).
For 0<a<4, f'(x)<0 for x<√a and f'(x)>0 for x>√a, so x=√a is a local minimum. The number of interior stationary points is therefore 0 for a=0 or a≥4, and 1 for 0<a<4.
2. Determine M(a)=max f(x) on [0,2] as a piecewise function. Check the manager's claim at a=1 and in general.
Read the worked answer
The interior stationary point, when it exists, is a minimum, so the maximum on [0,2] is at an endpoint. f(0)=1 and f(2)=9−6a. Compare: f(2)≥f(0) exactly when 9−6a≥1, i.e. a≤4/3.
Hence M(a)=9−6a for 0≤a≤4/3, and M(a)=1 for a≥4/3. At a=1, M(1)=3 at x=2, so the manager is right for this contract. But the claim that x=2 always maximizes is false: at a=2, M=1 at x=0. Also, raising a does not merely shift the curve down; the maximizing x changes, and after a=4/3 the maximum remains 1.
3. For a=2, compare f(0), f(1), and f(2), and identify the profit-maximizing production level. What does this show about the engineer's reply?
Read the worked answer
For a=2: f(0)=1, f(1)=1−6+1=−4, and f(2)=8−12+1=−3. The largest value is 1 at x=0, so the profit-maximizing choice is to set production to 0.
This confirms the engineer's warning: for sufficiently high supplier cost, shutting down can beat producing at x=2. It also contradicts the manager's claim that x=2 always maximizes profit.