Assignment 3 for each of the following give an example of a

Question

Assignment 3-

1. Suppose f(x) = 1 - √|x|cos(1/x) for x ≠ 0. If f(0) = 1, is f continuous at x = 0? Explain.

2. (a) Prove there is some real number x such that

x¹⁷⁹ + 163/√(1 + x²) = 119.

(b) Let f(x) = tan(x). Although f(π/4) = 1 and f(3π/4) = -1, there is no value of x ∈ [π/4, 3π/4] for which f(x) = 0. Why does this not contradict the Intermediate Value Theorem?

3. Suppose f(x) is a continuous function defined on [0, 1], and further suppose that 0 ≤ f(x) ≤ 1 for all x ∈ [0, 1]. Prove that there is some c ∈ [0, 1] such that f(c) = c. (Hint: Consider the function g(x) = f(x) - x).

4. Suppose g(x) is a continuous function, g(0) = 1 and lim_x→∞ g(x) = 0. Prove that there exists some real number c such that g(c) ≥ g(x) for all x ≥ 0.

5. For each of the following, give an example of a function that satisfies the desired properties, or prove that no such function exists.

(a) A function g that is the inverse of the function f, where f is defined on the interval [2, 3] by

f(x) = x² + x + 1.

(b) A polynomial p(x) such that |p(x)| has no minimum value.

6. Complete the proof of the inverse function theorem.