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Math Help - Mean Value Theorem True/False Questions

  1. #1
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    Mean Value Theorem True/False Questions

    Hey, can anyone tell me which of these are wrong?

    Hi, can anyone tell me which of these are wrong?:

    If f is differentiable on the set of real numbers and has two roots, then f' has at least one root. -- True

    If f is defined on the closed interval [a,b], differentiable on the open interval (a,b), and if f(a) = f(b) then there is a number c in (a,b) such that f '(c) = 0 -- True

    Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b), and let A = (a,f(a)) and B = (b,f(b)). Then there is atleast one point P on the graph of f where the tangent line is parallel to the secant line AB -- True

    Suppose f is differentiable on an interval I. If f is not injective on I, then there exists a point c in I such that f '(c) = 0 -- False

    If f '(x) = 0 for all x in an interval (a,b), then f(x) = 0 for all x in the interval (a,b) -- False

    Thanks!
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  2. #2
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    Quote Originally Posted by coldfire View Post
    Hey, can anyone tell me which of these are wrong?

    Hi, can anyone tell me which of these are wrong?:

    If f is differentiable on the set of real numbers and has two roots, then f' has at least one root. -- True

    If f is defined on the closed interval [a,b], differentiable on the open interval (a,b), and if f(a) = f(b) then there is a number c in (a,b) such that f '(c) = 0 -- True

    Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b), and let A = (a,f(a)) and B = (b,f(b)). Then there is atleast one point P on the graph of f where the tangent line is parallel to the secant line AB -- True

    Suppose f is differentiable on an interval I. If f is not injective on I, then there exists a point c in I such that f '(c) = 0 -- False

    If f '(x) = 0 for all x in an interval (a,b), then f(x) = 0 for all x in the interval (a,b) -- False

    Thanks!
    Not Sure if you wanted any detail so I will go ahead and give the results
    (i) false
    (ii) true
    (iii) fancier way of stating (ii) thus true
    (iv) again same statement as (ii) thus true
    (v) false, f(x) = constant (which could equal zero but not necessarily)

    if you require further clarrification or proofs please let me know and I will be more than happy to help
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