1. Giving f(x) = (x-2)³+1.

a). Show that the slope of the tangent line moving along the curve y=f(x), from left to right, is always upward (positive) except for a point, x=a,

b). Identify the single point, x=a, where the slope is not positive.

2. Assume the production cost (dollars/unit) for a unit of a certain product is given by f (x) = 2(1+x) e^-0.1x +100 where x>0 is the unit of products produced. (Note: For this problem, x can be considered a continuous variable. For instance, it is alright to have x=0.1, partial unit of the product. In a practical application, you can always round your final calculated result in a whole unit; e.g., x=0.85 as x=1 unit).

a). Show that the marginal cost of the production is a decreasing function of x (the marginal cost is simply the derivative of the production cost function, f. You can read the textbook (Sect. 2.1) for the definition of the marginal cost).

b). Determine the number of units that must be produced when the saving on marginal cost is no more than $0.05 per unit