Let the function h: R → R be bounded. Define the function f: R→ R by f(x) = 1+ 4x + x²h(x) for all x. Prove that f(0) = 1 and f'(0) = 4. (Note: There is no assumption about the differentiability of the function h.

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A: This is problem of Real Analysis, topology.

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A: Solution is given below:

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

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A: topic - linearization of a function

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