Solve question 37 completely 8:01 AM OVOLTE ile 4 17%*32.Recall, from Problem 14, that f 2 0 if f(x) 0 for all x in [a, b].(a) Give an example where f(x) 0 for all x, and f(x) 0 for some x inIf = 0.[а, b), and(b) Suppose f(x) 2 0 for all x in [a, b] and f is continuous at xo in [a, b]and f(xo) 0. Prove that °f 0. Hint: It suffices to find onelower sum L(f, P) which is positive.(c) Suppose f is integrable on [a, b] and f(x) 0 for all x in [a, b].Prove that "ƒ 0. Hint: You will need Problem 31 ; indeed thatwas one reason for including Problem 31.(a) Suppose that f is continuous on [a, b] and 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 /° fgtinuous 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)

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Solve question 37 completely 8:01 AM OVOLTE ile 4 17%*32.Recall, from Problem 14, that f 2 0 if f(x) > 0 for all x in [a, b].(a) Give an example where f(x) > 0 for all x, and f(x) > 0 for some x inIf = 0.[а, b), and(b) Suppose f(x) 2 0 for all x in [a, b] and f is continuous at xo in [a, b]and f(xo) > 0. Prove that °f > 0. Hint: It suffices to find onelower sum L(f, P) which is positive.(c) Suppose f is integrable on [a, b] and f(x) > 0 for all x in [a, b].Prove that "ƒ > 0. Hint: You will need Problem 31 ; indeed thatwas one reason for including Problem 31.(a) Suppose that f is continuous on [a, b] and 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 /° fgtinuous 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.*33.= 0 for those con-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 fl.(а) Prove that M; — m, < M; — т;.(b) Prove that if f 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 part" min(f, 0) are integrable on [a, b].38.Prove that if f is integrable on [a, b], thenHint: Thie fellewa

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We will be solving Q. 37 as mentioned.
Since you have posted a question with multiple subparts, we…