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Question: please solve all parts thanks...

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Please solve all parts! Thanks
1. (A). Use the intermediate value theorem to show that if fis continuous on [a, b and f(a) < 0, f(b) >0, then there existseE (a, b) such that f(c) = 0, ie., f has a root in (a, b). If, in addition, f(z) > 0, x E (a, b), use the Rolles theorem to show (by contradiction) that there is only one root in (a, b), .e., there cant be two or more roots (B). Use (A) to show that f(r) --1 has exactly one root in the interval (1,2) 2. Let f(x)- sin(2r) and o-0 (A). Caleulate the Taylor approximation Ts( (B). Use the Taylor theorem to show that (C). Write a Matlab program to compute the errors for - r for k-1, 2, 3, 4,5,6, and verify that | sin(2x) _ T,(z)| = 0(11-ro). (I will talk a little more about this at tomorrows class).
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