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Flashcards in S13 Deck (8)
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1

Let f : [a, b] → ℝ be a continuous function [a, b]. 

If ∫f(x) dx from a to b ≥ 0, then f(x) ≥ 0 for all x ∈ [a,b].

False 

Take for example :[−1,2]→ℝ such that f(x) = x.

Then

2

Let f : [a, b] → ℝ be a continuous function [a, b]. 

If f(x) < 0 for all x ∈ [a,b], then ∫f(x) dx from a to b

True 

By the mean value theorem, there exists c ∈]a, b[ such that

3

Let f : [a, b] → ℝ be a continuous function [a, b]. 

If ∫ |f(x)| dx from a to b = 0, then f(x) = 0 for all x ∈ [a,b].

 

True

(Too long to explain, man just check the answer sheet)

4

Let f : [a, b] → ℝ be a continuous function [a, b]. 

If c ∈ ]a,b[, then f(x) dx from a to c ≤ ∫ f(x) dx from a to b.

False

Take for example f :[−1,1]→ ℝ such that f(x) = −x.

Then

5

Let : [a, b] → R be continuous and F : [a, b] → R be a primitive of f on [a, b]. 

F is right differentiable at a.

True.

By the definition of primitive, F′(x) = f(x) for all x ∈]a,b[.

So lim F′(x) as x→a+ = lim f(x) as x→a+ = f(a).

We conclude by recalling the proposition that says that if lim F′(x) as x→a+ exists then F+′ (a) = lim F′(x). 

6

Let f : [a, b] → R be continuous and F : [a, b] → R be a primitive of f on [a, b]. 

If f(x) ≥ 0 for all x ∈ [a,b], then F is increasing.

True 

Follows directly from the fact that F′(x) ≥ 0 for all x ∈]a, b[.

7

Let f : [a, b] → R be continuous and F : [a, b] → R be a primitive of f on [a, b]. 

If f(x) ≤ 0 for all x ∈ [a,b], then F(x) ≤ 0 for all x ∈ [a,b]. 

False

Take for example f : [−1,0] → R defined by f(x) = x.

We have f(x) ≤ 0 and F(x)= 1/2*x2 > 0 for all x ∈ [−1,0[. 

8

Let f : [a, b] → R be continuous and F : [a, b] → R be a primitive of f on [a, b]. 

If F is injective and F(b) > F(a), then f(x) ≥ 0 for all x ∈ [a,b].

True

F is continuous and injective therefore monotone on [a,b]. Since F(b) > F(a), it is increasing and so f(x) = F′(x) ≥ 0 for all x ∈ [a,b].