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 that f 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.)