Suppose f:[a,b] → R is integrable, f(x) ≥ 0 for all X, and a f> 0. Which of the following is true? We must have f> 0 on [a, b]. There must be at least one point in [a, b] where f>0, but there need not be more than finitely many points where ƒ > 0. There need not be any points in [a,b] where f> 0. O There must be infinitely many points in [a,b] where f> 0, but it need not the case that f>0 everywhere.

Q: Let f : R R:f(x) = 10x + 7. Find the function g : R→R such that gof = fog = IR. %3D %3D

A: Given f: R -> R  such that f(x) = 10x + 7 and  g : R -> R such that g o f = f o g = IR

Q: Mathematicians B. Constante and M. Sosa are having an argument. Professor Sosa claims that it is…

A: Professor Constante is correct.

Q: If f is bounded and integrable on [a, b] such that | f(x) | 0).

A:

Q: Let f be continuous from [0, 1] to [0, 1]. By posing 6(x) = f(x) – x, show that fhas a fixed point…

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Q: Show that if f: A-> B is continuous and A is compavt then f(A) is also compact

A: Let , f : A →B be continuous function. Here , A is compact set. We need to prove that , f(A) is also…

Q: Let f: R³ R be the function defined as f(x, y, z)= √√₂² + y² + 2², 0. if (x, y, z) = Q³, otherwise.…

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Q: Show that if f:R → R is odd and the restriction of fto the interval [0, co) is strictly decreasing,…

A: The odd functions are functions that return their negative inverse when x is replaced with –x. This…

Q: Let f [0, 2] → R be a continuous function, with ƒ(0) ≤ f(1) ≤ f(2). Show that there : exists xo [0,…

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Q: (a) If ƒ and g are integrable, then gof is integrable. (b) If ƒ is increasing and g is integrable,…

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Q: [1+2p(x-1)+(x-1)² x≤1 x>1 [qx+p -) -) Find the values of q, in terms of p, for which fis continuous…

A: Given:f be a function defined by:a. To find the value of q in terms of p, for which f is…

Q: 44. Suppose that ƒ : [a, b] → [a, b] is continuous. Prove that there is at least one fixed point in…

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Q: Let f : [a, b] –→ R be twice differentiable and let f(a) = f(b) = 0 and f(c) > 0, where cE (a, b).…

A: Step:-1 Mean Value Theorem:-  Let a function f : a, b →ℝ be such that 1. f is continuous on a, b and…

Q: Suppose that f: A → R is continuous at a number a E A with f(a) > 0. Show that there exist numbers L…

A:

Q: True or false: If f: (0, 1)→ R is defined by f(x)=√x for every real number a in the open interval…

A: fx=x for all real xin the open interval  0,1

Q: Let f : D → R be defined by f(x) = (x+1)2/x for x ∈ D (D ⊆ R is the largest possible domain of the…

A: The domain of the function  f:   Domain: The domain of a function is the set of input or argument…

Q: Let a, b € R, a < b. Let f be an integrable function on [a,b), let c E (a, b), and let F(x) = |…

A: Solution for a: If f is differentiable at c, then F is differentiable at c Recall: First Fundamental…

Q: 3. Let f, g : [a, b] → [c, d] be cotinuous functions. Suppose that f is surjective. Prove that there…

A: We will use of intermediate value theorem to proof this

Q: 3. Let f(x, y) = (2x – y,x- 2y); (x, y) E R x R, (R is set of real numbers.) a) Show that f is one…

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Q: If f is differentiable on R and f(0) = 1,|f0(x)| < 1,∀x ∈ R. Show that |f(x)|≤|x|+ 1,∀x ∈ R.

A: Note that |f0(x)| &lt; 1 should have been |f'(x)| ≤ 1. Otherwise, the given inequality won't be…

Q: 11. Let f(x, y) = x² - y, g(t) = (Int) In(1-t), and h(x, y) = g(f(x, y)). Find the set on which h is…

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Q: the following statement. 'If yes, explain why. If no, provide a counterexample. For every real…

A: If the function is injective , then the above statement is true.

Q: Suppose that a, b R with a 0. Prove that [sz f≥ 0.

A:

Q: Suppose f and g are continuous on a closed interval [a, b] such that f(a) > g(a) and f(b) < g(b).…

A: I have used intermidiate value theoram of continuous function.

Q: Letf: R→ R be continuous. Suppose that f(1) = 4,f(3) = 1 and f(8) = 6. Which of the following MUST…

A: Solution :-

Q: Suppose that A,B⊂S such that Pr[A|B]=25, Pr[A]=12 and Pr[B′]=12. What is Pr[B|A′]?

A:

Q: Consider the function f : R³ → R defined as V(x1, x2, x3) E R³ f(x1, x2, X3) = X1X2+ X1X3 + X2X3 a.…

A: We are given the function f: ℝ3→ℝ defined as ∀x1, x2, x3∈ℝ3    fx1, x2, x3=x1x2+x1x3+x2x3. For (a),…

Q: Assume the function f(x, t) is continuous on the rectangle D = {(x, t) : a ≤ x ≤ b, c ≤ t ≤ d}.…

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Q: b) Let f: D→ C be a function and Y a subset of C. Show that f¹ [C-Y] =D-f¹ [Y].

A:

Q: How many functions f:R→R  exists such that : 2f(x)+2f(y)=f(x+y)+f(x−y)+2y holds for all real x and…

A: By choosing x=y=0. Also by choosing x=y we can see f(2x)=4f(x).  Moreover choose x=0 and use f(0)=0,…

Q: 4. Prove that the function f: [0, ∞)→ R; f(x) continuous. = √ is uniformly

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Q: Prove f(x) = x2+4 is not uniformly continuous on R.

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Q: If f is bounded and integrable on [a, b] such that | f(x) | 0).

A:

Q: B). For what values of r, reR is the function [(x+y+z) ƒ(x,y,z) = { x² + y² +z² 0, continuous on R.…

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Q: GeoGebrä Task 1: This rectangle has been decomposed along its diagonal. Recompose the two pieces to…

A: A rectangle is a closed quadrilateral with four sides and four angles. All the four angles of the…