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Mathematics, 22.07.2021 19:50 fmyear12
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) = β1, 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.β10< = 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 β 4x ^ 2 + 6, and g(x) = 3x ^ 3 β 8x. Prove, via the Intermediate Value Theorem, that the functions intersect at least twice between x = β2 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) β x.)
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