- #1

Swamifez

- 9

- 0

Prove that S = { n−1 |n ∈ N} is bounded above and that its supremumn

is equal to 1.

Use the Intermediate Value Theorem to show that the polynomial x4 +

x3 − 9 has at least two real roots.

Suppose that the function f is diﬀerentiable at a. Prove (without

quoting a theorem) that f 2 is diﬀerentiable at a.

Suppose that the function f is diﬀerentiable at a. Prove (without

quoting a theorem) that f 2 is diﬀerentiable at a.

Decide whether the following statements are true or false. Justify your

answers: proof or counterexample:

(a) Every continuous function f : [0, 1) → R which is bounded

takes on its maximum

(b) There exists a function f : [−1, 1] → [−1, 1] with no x ∈ [−1, 1]

satisfying f (x) = x.

If anyone knows anything on any of these problems, its highly appreciated!