Question

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.)

Answer 1

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 true that these exist a differentiable function FR IR such that I'= f. Counter Exemple. The following function is Riemann integrable but there does not exist my F: IR→IR St F'=f if -exco if ocaç f (x) = 1 (b) Consider the function g(x) = f(a)-* From the question 910) = f(o)-0 =0 Also g'(x) = f(x)-1 >0 (os f'(x)>, 1 x GIR). Hence, g(x) is on increasing function. Since g(0) = 0, 9(x) 7,0 f(x) x 70 x 70 = f(x) x x &xxo we get &xxo (proved) 13 (C) Given f(x) = x + x Since s'(x) = 13 x ² + 1 & XEIR, f(x) is a continuous strictly function. Since, every Strictly increasing continuous funch on has on inverse s exist. increasing Next Let & (1) = x

1 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) = x + x f(0) = 0 = f'(o)=0 Therefore, f' I f (x) dx 1.5 (1) - 0.0 - ) f(x) dx sto S (x² + x) dx O 2 11 - [ + 2 0 14 22 (Answer) 2.