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Math Help - Analysis (MVT)

  1. #1
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    Analysis (MVT)

    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 starred bit I can't do. I'm getting so frustrated. Any hints would be appreciated. Thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Aidan1930 View Post
    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 starred bit I can't do. I'm getting so frustrated. Any hints would be appreciated. Thanks
    Clearly \varphi'(x_0)=\lim_{x\to x_0}\frac{\varphi(x)-\varphi(x_0)}{x-x_0}=?

    It is clear for x=0 since \varphi'(0)=\lim_{x\to0}\frac{\varphi(x)-f'(0)}{x}=\lim_{x\to 0}\frac{f(x)-f(0)-x\cdot f'(0)}{x^2} and since f is differentiable we are allowed to use L'hopitals rule to get \lim_{x\to0}\frac{f'(x)-f'(0)}{2x}=\frac{f''(0)}{2}.

    Try the rest yourself.
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