LTE 8:01 AM Ø VOLTE il: 4 17% *32. Recall, from Problem 14, that j 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 in [a, b], and f (b) Suppose f(x) 2 0 for all x in [a, b] and ƒ is continuous at x, in [a, b] and f(xo) > 0. Prove that f> 0. Hint: It suffices to find one = 0. lower 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 ["f > 0. Hint: You will need Problem 31; indeed that was one reason for including Problem 31. *33. (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; there is 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 conditions g(a) = g(b) = 0. Prove that f = 0. (This innocent looking fact is an important lemma in the calculus of variations; see the Suggested Reading for references.) Hint: Derive a contradiction from the assumption f(xo) > 0 or f(xo) < 0; the g you pick will depend on the behavior of ƒ 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 = that f is integrable on [0, 1] and that 'f = 0. (Every lower sum is p/q in lowest terms. Show
LTE 8:01 AM Ø VOLTE il: 4 17% *32. Recall, from Problem 14, that j 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 in [a, b], and f (b) Suppose f(x) 2 0 for all x in [a, b] and ƒ is continuous at x, in [a, b] and f(xo) > 0. Prove that f> 0. Hint: It suffices to find one = 0. lower 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 ["f > 0. Hint: You will need Problem 31; indeed that was one reason for including Problem 31. *33. (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; there is 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 conditions g(a) = g(b) = 0. Prove that f = 0. (This innocent looking fact is an important lemma in the calculus of variations; see the Suggested Reading for references.) Hint: Derive a contradiction from the assumption f(xo) > 0 or f(xo) < 0; the g you pick will depend on the behavior of ƒ 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 = that f is integrable on [0, 1] and that 'f = 0. (Every lower sum is p/q in lowest terms. Show
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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