The Mean Value Theorem: Total 10 points You should consider the parts of Question 7 as coming together to ask about a single mathematical topic. If you choose this question, do it all. By missing any part of this question, you seriously undermine your credibility on the parts you have answered... [a] [1 point] lIll-liven o 4:: h and that f is a function dened on the interval [-11, In], what does Fermat's theorem say about the behavior of the function f at a given number .12 that lies between a and b? {b} [I point] Given that o. a: b and that f is continuous at every number in the interval [.1115], what fundamental fact do you know about the function f; a fact that dapends upon the completeness of the real number system? 4 [c] [I point] Statement of Rolle's Theorem: We suppose that o e: t and that f is oontinnous at every numb-er in the interval [e1 1?] and that the derivative ts} of f exists at every number s: between a and t and that f {a} = f (tr) 'What does Holle's theorem now tell us? Do not write any proof in this part of the question! {cl} [3 points] Proof of Rolle's Theorem: Referring to what you wrote in part (c), write a careful explanation of whyr Rolle's theorem is true. (You will use what you wrote in parts {a} and (b) and {e}. So don"t even attempt this part unless you these earlier parts (a) and (is) properly.) {e} [I point] Statement of the Mean Value Theorem: We suppose that e -=:: b and that f is continuous on and interval [mt] and that the derivative 1\" [33} of f exists at every number 3: between :1 and b. \"That does the mean value theorem now tell us? Do not write any proof in this part of the question! (f) [3 points] Proof of the Mean 1|infaiue Theorem: Referring to what you wrote in part [f], write a careful explanation of why the mean value theorem is true