MA104 EXAM IIINOTE: Make sure to introduce any necessary hypotheses that are not explicitlystated in problems. Consider this part of your task when asked to define things.For example, to state that a function is Darboux integrable, we must first assumeit is bounded, because otherwise M(f; S) will not have meaning.12Problem 1:Define carefully the phrases:(i) “f is differentiable at x0.”(ii)“x0is a local extremum for f .”Supp osing both statements are true, what can we conclude? Formulate a precisestatement, and prove your claim. Does the converse to your statement hold?3Problem 2:State precisely Rolle’s Theorem (RT) and the Mean Value Theorem (MVT). Sketchan example of each.Using Problem 1, prove Rolle’s Theorem.4Problem 3:As you are walking through Evans, you overhear a freshman say“I can’t believe I lost points because I forgot the +C. Why does the antiderivativealways have to have +C?”State a mathematically precise answer to this question, and prove it. In otherwords, if two functions have the same derivative on an interval (a, b), what can weconclude? Does the converse to your statement hold?5Problem 4:Define precisely the phrase “f is Darboux integrable on [a, b ].” (There are a lot ofauxiliary definitions behind this, like partition, M (f ; S), U (f ; P ), L(f), etc.)Prove the key lemma L(f ; P ) ≤ U(f; Q), for any partitions P, Q. Why is it a keylemma; in other words, what does it show is ‘well defined’?6Problem 5:State both versions of the Fundamental Theorem of C alculus. Prove one of them.You may use the Riemann definition of integral, but if so, briefly indicate how itdiffers from the Darboux integral.7Problem 6:Prove that, if f : [a, b] → R is continuous, then it is integrable, and furthermorethat there exists some c ∈ (a, b) such thatf(c) =1b − aZbaf.(The second part of the problem is a theorem in the book, but we did not do it