Please do 3 and 4 Please please please 1. Let f : [1,4] → R be defined by f(x) = 4x² for 0 ≤ x ≤ 2 and f(x) = 2x for2 < x≤ 4. Let P = {-1,0, 2, 3, 4}. Find L(f, P) and U (f, P).2. Let f: [0, 1] → R be defined by f(x) = -8. Prove, directly from the definition, thatf is integrable.3. Let f: [0, 1] → R with f(x) = 0 if x is rational and f(x) = -4 if x is irrational. Provewhether or not f is integrable.4. Let f: [a, b] → R be integrable and let c> 0. Use the definition OR the Archimedes-Riemann Theorem to prove directly that cf is integrable and that-b["(of)" (ef) = c f². fa

Given : f :0 , 1→ℝ : fx=0 , x is rational-4 , x is irrational To Check : f is…

Answered: 3. Let ƒ : [0, 1] → R with f(x) = 0 if…

3. Let ƒ : [0, 1] → R with f(x) = 0 if x is rational and f(x) = −4 if x is irrational. Prove whether or not f is integrable.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter9: Multivariable Calculus

Section9.3: Maxima And Minima

Problem 33E

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Question

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Please do 3 and 4 Please please please

1. Let f : [1,4] → R be defined by f(x) = 4x² for 0 ≤ x ≤ 2 and f(x) = 2x for
2 < x≤ 4. Let P = {-1,0, 2, 3, 4}. Find L(f, P) and U (f, P).
2. Let f: [0, 1] → R be defined by f(x) = -8. Prove, directly from the definition, that
f is integrable.
3. Let f: [0, 1] → R with f(x) = 0 if x is rational and f(x) = -4 if x is irrational. Prove
whether or not f is integrable.
4. Let f: [a, b] → R be integrable and let c> 0. Use the definition OR the Archimedes-
Riemann Theorem to prove directly that cf is integrable and that
-b
["(of)
" (ef) = c f². f
a

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