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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Originally Posted by coldfire

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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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