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3. Let f: R+R be a function. (a) Assume that f is Riemann integrable on [a,...


3. Let f: R+R be a function. (a) Assume that f is Riemann integrable on [a, b] by some a < b in R. Does there always exist a
3. Let f: R+R be a function. (a) Assume that f is Riemann integrable on [a, b] by some a < b in R. Does there always exist a differentiable function F:RR such that F' = f? Provide either a counterexample or a proof. (b) Assume that f is differentiable, f'(x) > 1 for every x ER, f(0) = 0. Show that f(x) > x for every x > 0. (c) Assume that f(x) = 2:13 + x. Show that f-1 exists in other words, that f is invertible). Given that f-l(1) = a, compute the value of [ s-'(x)dır. (Your answer may be given in terms a.)
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Solution Given f. 1R IR be a function. some (a) let f be Riemamm integrable function on [a,b], by acb. It is not always true1 To evaluate the value of 5 (odx, we will use the following formula 1 f (x) dx b 5(b) – of (a) - ) f(x) dx 13 1 Since, f(x)

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3. Let f: R+R be a function. (a) Assume that f is Riemann integrable on [a,...
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