Suppose that f has continuous derivative and is twice differentiable in R. Suppose that f(a) = f(b) = 0. Show that for every x E (a,b), there is § E (a,b) such that f"(€) (æ – a)(x – b). f(x) = 2

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A: Explained below

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A: Uniformly Continuous:-  Here we prove the result using a theorem of uniformly continuous.

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A: Yes, the sequence is given in step 2.

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A:

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A: a= 8.   See the attachment for the complete solution.

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A: We will find the derivative using Fundamental Theorem of Calculus.

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A: f(x) will be continuous at x = 0 if LHL = f(0) = RHL

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A: It is given that ∫13fxdx=0 and -1≤fx≤1.

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A: We have to find f(c)

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A: In the given question, the concept of the Fundamental Theorem of Calculus is applied. Fundamental…

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A: Given that:                           f(x) is continuous in [0, 9].…

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A:

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A: This question can be solved using properties of definite integrals.

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A:

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A: We have to find

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A: Follow the procedure given below.

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A:

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A: Every differentiable function is continuous.

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A: Explanation of the solution is given below...