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Let f and g be twicedifferentiable realvalued functioned defined on R. If f'(x) > g'(x) for all x > 0, which of the following inequalities must be true for all x > 0 ?
(A) f(x) > g(x)
(B) f''(x) > g''(x)
(C) f(x)  f(0) > g(x)  g(0)
(D) f'(x)  f'(0) > g'(x)  g'(0)
(E) f''(x)  f''(0) > g''(x)  g''(0)
 one year ago
 one year ago
Let f and g be twicedifferentiable realvalued functioned defined on R. If f'(x) > g'(x) for all x > 0, which of the following inequalities must be true for all x > 0 ? (A) f(x) > g(x) (B) f''(x) > g''(x) (C) f(x)  f(0) > g(x)  g(0) (D) f'(x)  f'(0) > g'(x)  g'(0) (E) f''(x)  f''(0) > g''(x)  g''(0)
 one year ago
 one year ago

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aacehmBest ResponseYou've already chosen the best response.1
This is an interesting question, have you gotten anywhere with it?
 one year ago

aacehmBest ResponseYou've already chosen the best response.1
I'd say CE are ruled out, since you can't make any assumptions about f(0) or g(0)
 one year ago

aacehmBest ResponseYou've already chosen the best response.1
Actually, nevermind, you can. Because if f'(x) > g'(x) very close to zero, the definition of differentiability seems like it would guarantee that f'(0) > g'(0)
 one year ago

LogicalAppleBest ResponseYou've already chosen the best response.1
I have not looked at the problem yet myself. I posted this as a challenge. The question is saying that, after x = 0, f increases faster than g. We certainly can eliminate A because this says nothing of the values of f(x) or g(x). I am learning towards C only because of what I imagine the function does at the point x = 0. Taking into account your previous post, it appears we are in agreement.
 one year ago

aacehmBest ResponseYou've already chosen the best response.1
It's hard to think of two functions that don't adhere to A though
 one year ago

LogicalAppleBest ResponseYou've already chosen the best response.1
f(x) = x^3  10 g(x) = x^2 Here, f(1) < g(1) so A fails. But f'(x) > g'(x) for all x > 0
 one year ago

aacehmBest ResponseYou've already chosen the best response.1
yeah, I guess you're right, so are we going with C or D? Because if it's twice differentiable, then D sounds like a better answer to me
 one year ago

LogicalAppleBest ResponseYou've already chosen the best response.1
C says f(x)  f(0) > g(x)  g(0) D says f'(x)  f'(0) > g'(x)  g'(0) given: f'(x) > g'(x), f'(x)  g'(x) > 0 We can write this as a composite function (f  g)'(x) > 0 This means that the composite function (fg)(x) is increasing. Which equivalently means f(x)  g(x) > 0 This is why I think the answer is C. The logic isn't sound but the line of reasoning seems ok.
 one year ago
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