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Calculus Proofs Help

  1. Nov 12, 2006 #1
    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!
     
  2. jcsd
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