Question

Please help! Not understanding Calculus question! Will give medals!

Answer 1

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.)