1.Suppose C(x) is a function representing the cost (in dollars) of producing x units of energy, and R(x) is a function representing the revenue (in dollars) of selling x units of energy. Suppose further that both functions are continuous for all x > = 0. A. Is there necessarily some x value, let's call it C, that will maximize profit over all x > =0? Explain your answer.

B. Suppose we know we can't produce more than 1,000 units of energy. Is there necessarily some x-value, call it C, that will maximize profit over the interval [0,1000]? Explain your answer.

C. Again, assume that we can't produce more than 1,000 units of energy. Is there necessarily some x value, let's call it C, for which the profit is exactly 0? Explain your answer.

2.

A. Produce a function f(x) that satisfies the following conditions:

I. Its domain is all real numbers.

II. It has no maximum and no minimum on the interval [ 1,3] .

III. It satisfies f(1) = 1 and f(3) = –1, but there does not exist a c between 1 and 3 such that f(x) = 0.

B. Construct a function f(x) that satisfies the following conditions:

I. f is continuous for all x.

II.–10< = f(x) < = 10 for all x.

III. f does not have a maximum or minimum on the interval [negative infinity, infinity].

3.Suppose f(x) = x ^ 4 – 4x ^ 2 + 6, and g(x) = 3x ^ 3 – 8x. Prove, via the Intermediate Value Theorem, that the functions intersect at least twice between x = –2 and x = 4.

4.Suppose f(x) is continuous on the interval [0,1] with 0< = f(x)< = 1 for all x. Prove thatf has a fixed point; that is, there exists a c in the interval [0,1] such that f(c) = c. (Hint:You need to use the Intermediate Value Theorem on the function h(x) = f(x) – x.)