Assume that f and g are differentiable functions such that f(0) 2 g(0) and that for all r€R, f'(z) > g(x). Prove that f(z) > g(z) for all >0.

Q: Show there does not exist a continuously differentiable function f on [O, 2] such that f(0) = -1,…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Define g : R →R x sin (1/x), g(x) = 10, x # 0 x = 0. Explain why g is continuous at 0; in…

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Q: consider the function f defined on [0,∞),f(x)=(x^r)sin(1/x),for x≠0 and f(x)=0 and for x=0, where r…

A: Solve the following

Q: Let the function ƒ : R → R be given by 1- cos(x) if x + 0, f(x) := if x = 0. %3D 1 Prove that f is…

A: The solution is given as

Q: Let f be a differentiable function satisfying the condition: f(x) for all x, y (#0) eR and f(y) 0.…

A: In this we have given a functional relations and from that we have to find out the function f(x), so…

Q: Let f : R → (−1, 1) be given by f(x) = x/(1 + |x|). Show that f is differentiable at x = 0, with…

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Q: Let f and g represent differentiable functions such that f ″ ≠ 0 and g″ ≠ 0. Show that if f and g…

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Q: 2. f(x) = Vx – 1 at x = 5

A: Function is continuous and differentiable both at x= 5.....for detail...go through the solution

Q: B) Show that fx) = sin for x * 0. for x= 0. X = is not continuous.

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Q: What is the minimum f(x) value of f(x) = sin x on the interval [0, 2]? %3!

A: Derivative Test: find values of x for which f'(x)=0 Then find f''(x) and put values pf x from above…

Q: Define a function f : R² → R² by f 2.у 1. Show that f is differentiable on A = E R² |x > 0, y > 0}…

A: To show that the function f defined as below is differentiable on A and compute its derivative

Q: f: R (0, ) be differentiable function satisfying f(f(x)) = f'(x) for each 'x. Prove that no such…

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Q: (b) Test the continuity of the function f(x) = sgn(lxl) at x = 0. Is f(x) %3D differentiable at x 0?…

A: The function f(x) = (sgn|x|) To test the continuity of the function at x = 0  Also to determine…

Q: Assume that f : (0, 1) → R is a differentiable function with the derivative f' increasing on (0,…

A: Given f:0,1→R is a differentiable function and derivative of f is increasing in 0,1 that is f'x>0…

Q: Determine if the function is continuous at point (0,0): F(x, y)=**y°cos(x*+y³) (x²+y²)2

A: Given that Now take x=0, Now take y=0,

Q: Suppose that e(x) and f(x) are differentiable on [0, 1] and that 0<|e'(x)| ≤ |f'(x)| for all x ∈ (0,…

A: Given: ex and fx are differentiable on 0,1 and that 0&lt;e'x≤f'x, ∀x∈0,1. To Show: ex-ey≤fx-fy,…

Q: Suppose that F(t) is a twice-differentiable function of one variable. Define the function f(x,y) =…

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Q: Suppose f(x), f;(x), g,(x), and g,(x) are differentiable functions. Show that, if W = | f,(x) f,(x)|…

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Q: Find the derivative of f(x) at r = 0 using the definition of the derivative as a limit. %3D Va +I

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Q: Let f(x, y,z) = z-sin(xy), x t, y Int, z et1. Find derivative of f directly and by using chain rule…

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Q: Let h(x) = f(g(x)), where f and g are differentiable on theirdomains. If g(1) = 3 and g′(1) = 5,…

A: Given: Let h(x) = f(g(x)), where f and g are differentiable on theirdomains. If g(1) = 3 and g′(1) =…

Q: Show that the function f defined by (x, y) = (1, –1) f(r, y) = { x2 + y (x, y) # (1,–1) r + y is not…

A: To show that the function f defined by fx, y=1, x, y=1, -1x2+yx+y, x, y≠1, -1is not continuous at 1,…

Q: (2) Are all entire functions differentiable on D?

A: Here we check whether given statement is true or false. Are all entire functions differentiable in…

Q: Determine if the function is continuous at point (0,0): F (x, y) _x³y°cos(x³+y³) (x²+y²)²

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Q: 2. Determine the points where f is continuous where x + Y if (x, y) # (0,0) f(x, y) = x2 + y2 if (x,…

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Q: Let f and g be continuous on the closed interval [0, 5]. Suppose that J f(x)dx = A, f(x)dx = B, and…

A: Given: ∫05f(x)dx=A∫03f(x)dx=B∫53g(x)dx=C…

Q: ery differentiable function y=f(x) defined

A: Given statement is: Every differentiable function y=f(x) defined on an open interval (a,b) has an…

Q: Show that the function f defined by (r, y) = (1, –1) (r, y) # (1, –1) f(r, y) = 1² + y is not…

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Q: 3) Examine the limit of 1 (x² +y²).sin- x*+ (x, y) (0,0) f(x, y) = (0, (x, y)=(0,0)

A: Solution:Givenf(x,y)=x2+y2 sin 1x2+y2To Eamine the limit of a functionFormula:limx→0 sin xx=1

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

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Q: Determine where f(x) = - is differentiable. sin xcos x пл 2in = 0, ± 1, ±2, .. Ox = na, n = 0, ± 1,…

A: Given function is: fx=7sinxcosx To find the points where the function is differentiable.

Q: Show that f(z) =xy+iy is everywhere continuous but is not analytic

A: Answer

Q: Find a positive value, c, for x, thot satisfies the Mean valve Theorem for derivatives for…

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Q: Show that the function is continuous over ]-infinity, +infinity[ 4 x* sin(1/x) si x#0 f(x)=: si x=0

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Q: Determine if a function f : R2 → R defined by f (x; y) = { x sin(1/x2+y2) , if (x, y) ≠ (0,0) and 0…

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Q: Show that the function f(z) = sin x cosh y + i cos x sinh y is continuous as well as analytic every…

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Q: if x = 0 f (x) = %3D x* sin(-) otherwise

A: In the given question, concept of continuity is applicable. Continuity Continuity for a graph y=fx…

Q: Q\ Use appropriate functions to guarantee that (). x sin if x # 0 h(x) = 0, if x = 0 is continuous…

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Q: Let f(x) E R[x]. Suppose that f(a) = 0 but f' (a) + 0, where f' (x) is the derivative of f(x). Show…

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Q: Let f(x, y) = x2y4 sin(1/ (sqrt(x2+y2)) where (x, y) cannot be (0, 0) and f(0, 0) = 0 Is f…

A: Answer

Q: A bounded, 3B > 0 such that |f(x)| < B Vx € (0, 1), continuous function on (0, 1) with unded…

A: We will find an example

Q: Consider the function f defined on [0,∞), f(x)=(x^r)sin(1/x), for x≠0 and f(x)= 0, where r>0.…

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Q: Let Assume that f has a continuous derivative on a, b. Show tha p(x) such that |f(r) – p(x)| 0. (b)…

A: Given:- f has a continuous derivative on [a , b ]. p(x)such that f(x)-p(x)&lt;ε and f'(x)-p'(x)&lt;ε…

Q: Suppose that f(x)=xu(x), where u(x) is a differentiable function at x=2 such that u(2)=3, u'(2)=1.…

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Q: Show that the function f(z) = z = is differentiable but not analytic at z = 0

A: Givenf(z)=zz

Q: Show that there is no differentiable function f on [0, 2] with f (0) = -4, f (2) = 4 and %3D f' (x)…

A: Given: A function f such that f(0)=-4,f(2)=4 and f'(x)≤3, ∀x∈[0,2] According to Mean Value Theorem,…

Q: Let f and g be continuous on the closed interval [0, 5]. Suppose that | f(x)dr = A, | f(x)dx = B,…

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Q: Let f and g be continuous on the closed interval [0, 5]. Suppose that | f(æ)dx = A, f(x)dx = B…

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Q: B) Show that f(x) sin for x * 0. for x= 0. is not continuous.

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Q: Assume that f(x) and g(x) are differentiable for all x. Let h(x) = x* f(x), find h'(x). h'(x) = help…

A: We will use the product rule