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#1 2010-02-26 09:53:54
- AlexGo
- Guest
Analysis
This question is driving me mad.
Let f be continuous on [−1, 1] and twice differentiable on (−1, 1). Let φ(x) = (f (x) − f (0))/x for x = 0 and φ(0) = f ′ (0). Show that φ is continuous on [−1, 1] and differentiable on (−1, 1). Using a second order mean value theorem for f, show that φ′(x) = f′′(θx)/2 for some 0 < θ < 1 . Hence prove that there exists c ∈ (−1, 1) with f ′′ (c) = f (−1) + f (1) − 2f (0).
It's the bit in bold I can't do. I'm getting so frustrated. Any hints would be appreciated. Thanks
#2 2010-02-26 23:16:14
- AlexGo
- Guest
Re: Analysis
Bump.
Ricky?
#3 2010-02-27 08:22:17
- Ricky
- Moderator
- Registered: 2005-12-04
- Posts: 3,791
Re: Analysis
You're having trouble because the proposition is false. Let f = e^x, then
Which certainly does not fit the conclusion.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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