Solve question 38 completely LTE8:02 AM ØVOLTE L C 4 17%*33.(a) Suppose that f is continuous on [a, b] and l fg = 0 for all con-tinuous functions g on [a, b]. Prove that f = 0. (This is easy; thereis an obvious g to choose.)(b) Suppose f is continuous on [a, b] and that /fg = 0 for those con-tinuous functions g on [a, b] which satisfy the extra conditionsg(a) = g(b) = 0. Prove that ƒ = 0. (This innocent looking fact isan important lemma in the calculus of variations; see the SuggestedReading for references.) Hint: Derive a contradiction from theassumption f(xo) > 0 or f(xo) < 0; the g you pick will depend onthe behavior of f near xo.34. Let f(x) = x for x rational and f(x) = 0 for x irrational.(a) Compute L(f, P) for all partitions P of [0, 1].(b) Find inf {U(f, P): P a partition of [0, 1]}.*35. Let f(x) = 0 for irrational x, and 1/q if x = p/q in lowest terms. Showthat f is integrable on [0, 1] and that f = 0. (Every lower sum isclearly 0; you must figure out how to make upper sums small.)*36. Find two functions f and g which are integrable, but whose compositiongof is not. Hint: Problem 35 is relevant.*37.Let f be a bounded function on [a, b) and let P be a partition of [a, b].Let M; and m; have their usual meanings, and let M,' and m' have thecorresponding meanings for the function |f].(a) Prove that M,' – m²' < M¡ – mị.(b) Prove that if ƒ is integrable on [a, b], then so is [f].(c) Prove that if f and g are integrable on [a, b], then so are max(f, g)and min(f, g).(d) Prove that f is integrable on [a, b] if and only if its "positive part"max(f, 0) and its “negative partť" min(f, 0) are integrable on [a, b].38.Prove that if f is integrable on[a, b], thenHint: This follows easily from a certain string of inequalities; Problem1-14 is relevant.*39. Suppose f and g are integrable on [a, b] and g(x) 2 0 for all x in [a, b].Let P be a partition of [a, b]. Let M, and m' denote the appropriatesup's and inf's for f, define M;" and m;" similarly for g, and define M;and m¡ similarly for fg.(a) Prove that M; < M¿'M;" and m; 2 m;'m;".(b) Show thatU(fg, P) – L(fg, P) < ) [M¿'M;" – m{m;"](t; – i-1).13. Integrals 26.

We will use concepts of Riemann integration

Hint: This follows easily from a certain string of inequalities; E 1-14 is relevant. *39. Suppose ƒ and g are integrable on [a, b] and g(x) 2 0 for all x is Let P be a partition of [a, b]. Let M{ and m,' denote the appi sup’s and inf's for f, define M" and m;" similarly for g, and de and m¡ similarly for fg.