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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Originally Posted by Aidan1930

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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Clearly

It is clear for since and since is differentiable we are allowed to use L'hopitals rule to get .

Try the rest yourself.