Let f and g be continuous functions defined on a compact interval [a, b]. Assume that g(x) > 0 for all x E [a, b]. Prove that there exists xo E [a, b] such that qu | f(x)g(x)dx = f(wo) / g(x)dx. a (Hint: consider using change of variable and the integral mean value theorem or the second mean value theorem.) Is the same conclusion true if we remove the assumption g(x) > 0? If yes, please give a proof; if no, please give a counterexample.

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

A:

Q: Prove the following statement: Assume f is a continuous function on [a, b], k is any real number |…

A:

Q: Suppose that f(x) is a continuous function on [a , b] and F(x)= | f(t) dt a Part 1 of the…

A:

Q: Prove that if f is a continuous function on a closed interval [a, b], then

A: We have to prove the given inequality for a continuous function on a closed interval [a,b].

Q: Let X be Hausdorff and f : X → X be a continuous function. Show that {x : f (x) = x} is closed in X.

A: Let, .We show  is open and so F is closed.

Q: 2. Determine if a function f : [0 ; 2pie] → R defined by f(x) = sin 2pie is continuous

A: Determine if a function f : [0 ; 2pie] → R defined by f(x) = sin 2pie is continuous.

Q: Suppose that f is a continuous function on the interval [0,3] such that f(0) 4. Apply the…

A:

Q: pose that the function f is continuous everywhere. Suppose that F is any antiderivative of f, and…

A: In this question, the concept of Integral is applied. Integration Integration is a method of…

Q: Let f(x) = |x|+|x−2], x € R. Find points where f is not differentiable.

A:

Q: Assume that f is continuous on the set of real numbers. Find a and b given that cb 3 f(a)da - [³…

A:

Q: Check the continuity of the function f(x). Explain why f(x) is continuous or not. x+1 =,x x2X > 1

A:

Q: Suppose that the function f: [0, 1]  in R is continuous and thatf (x)>= 2     if 0 <= x <…

A:

Q: Prove that for any function f : X → Y and for any subsets Y₁ and Y₂ of Y, ƒ¯¹(Y₁ NY₂) = ƒ¯¹(Y₁) Ñ…

A:

Q: 2. Determine if a function f : [0, 27] → R defined by f(x) = sin 2n is continuous.

A:

Q: Consider the PIECE-WISE function: Ат-В T-2 if r<1, f(x) = { 32, if 1<x < 2, Ba? — А, if a 22.

A:

Q: Prove and show complete solution: Let ƒ be defined on an open interval (a, b) and assume that ƒ has…

A:

Q: q17. Let f, g, h be continuous functions on an interval [a, b]. If f is orthogonal to g and to h,…

A:

Q: Let f: X --> Y be a function, and let U be a subset of X.  (1) Prove or disprove that f maps X \ U…

A:

Q: Let f: IR be a function. (a) Prove that if f is differentiable at some c I for which f(c) ‡ 0, then…

A:

Q: Assume that f is continuous on the set of real numbers. Find a and b given that b 3 [ f(x) dx - f₁…

A:

Q: Determine the set of points domains at which the function is continuous.  G(x,y)= 2x tan y

A: To Find: Set of points in domain at which the function 2x tany is continuous.

Q: Assume that f and g are differentiable functions such that f (0) > g(0) and that for all x E R,…

A:

Q: 4. Let f:R→ R be a function defined by f(x)=x for every xeX . Show that f is a continuous.

A: Let A ⊆ R , let f:A → R and let c ∈ A. We say that f is continuous at c if, given any number ε&gt;0,…

Q: If f(z) is an analytic function of z and f'(z) is continuous at each point within and on a closed…

A: We will use greens theorem and properties of an analytic function.

Q: Let f(x) = cos (x). Prove that f is continuous on [a,b] and f has a derivative in (a,b), for all…

A: Continuity: A real-valued function fx is said to be continuous at a point x=a if for every ε&gt;0…