- #1
Swamifez
- 9
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Hey, I need help with a couple of questions in my analysis calc and proof class. Thanks in advance!
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!
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!