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 y?

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: Find the values of a and b that make f(x) continuous for all real x. be* + a + 1, ax² + b(x +3), a…

A: The function f(x) is said to be continuous if the left-hand limit is equal to the right-hand limit…

Q: Let f(z) - 4z-1 pnd alz) - ar+b, where a pnd b are positive constants. If the composite

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Q: Find the particular antiderivative F(x) of f(x) = x6 1 that satisfies F(1) = 4. - %3D

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Q: A function (f) is homogeneous of degree • if and only if : ôf + y = Af (x, y). Does the function…

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Q: of and ду of For the function f(x,y) = In (x+ 3y), find

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Q: f(x)={x^2−9x≤c, 2x−10 x>c } i

A: Consider function f(x)=x2-9           x≤c2x-10        x>c Now  limx→c+ f(x)=limx→c+…

Q: Q2: Expand f(x,y) = [sin (x y³)]² about (5, 1).

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Q: Is there a continuous function such that - f: [1,2] U [3,4] → R - f(1) * f(4) < 0 - f(x) not egual 0…

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Q: x - y -2x Prove by definition that for f(r, y) fy(x, y) = %3D x+ y' (x+ y)2

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Q: 4) Determine the function F(x)=[ƒ(1)dt, where ƒ(x)=sgn(x-x²) for xe[-1,2]. -1

A: f(x) is discontinuous but F(x) is continuous

Q: Let f(x) = x²+x+ 2 = Z₂ (x) prove that f(x) irreduciable Poly

A: The given function is  f(x)=x2+x+2.

Q: Show that f (x) = 1 − |x| satisfies the conclusion of the MVT on [a, b] if both a and b are positive…

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Q: For what value of b is

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Q: 1. Find of ахду OF f(x,y) = xy (x+y-1) -x2

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Q: Consider the function xy f(x, y) x²+xy+y² k if (x, y) = (0,0) if (x, y) = (0,0) State, with…

A: Given :  f(x, y) = xyx2+xy+y2if (x, y)≠(0, 0)kif (x, y) = (0, 0) To find : The value of k if it…

Q: Let g(x, y) = x4 + y² – 4x³ + 5x – 6y+3. Show that g(x,y) continuous (actually at least aC')…

A: Recall [ Implicit Function Theorem ]

Q: Find and classify all critical points for the function f(x, y) = x + y – 4xy + 2 %3D

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Q: Show that f(x1, x2) = 2 - (x1 - 1)^2 - e^x2^2 is concave.

A: the given function is fx1,x2=2-x1-12-ex22 To find the function is concave we have to show all its…

Q: of of and ду For the function f(x,y) = In (7x + 9y), find

A: The given function,     f(x,y)=ln7x+9y

Q: of Let f(x,y) = xy° +7. Evaluate at (x,y) = (-7,1) and interpret the results. ду -

A: In this question we have find δfδy at (x, y)=(-7, 1)

Q: 7. Find a and b such that f is differentiable everywhere. f(x) = {+b, S ax³, lx² + b, x 4

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Q: Determine whether the function 2х + 1 f(x) %3D 2r2 2x2 – x – 1 is continuous: a. at 2; b. at 1.

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Q: what is the antiderivative F(x) of f(X) = 3x1/2-x-1 that satisfies the condition  F(4)= 0? A,…

A: The given function is fx=3x12-x-1 that satisfies the condition F4=0. Find the antiderivative of Fx.

Q: 4. Give an example of a continuous function f : (0, 1) → R that cannot be uniformly approximated by…

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Q: Suppose that ƒ(x) = x2 and g(x) = | x | . Then thecompositions( ƒ ∘ g)(x) = | x | 2 = x2 and (g ∘…

A: Given: The composite of the two functions are given as,

Q: Suppose that f : R →R is continuous and that its image f(R) is bounded. Prove that there is a…

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Q: Use the comparison theorem with f(x) = = 1 √x r∞ 1 •√₂ √I-1 to show that is divergent.

A: Solution is given below:

Q: Find aland U so that the function S 7z³ – 3x² + 7, z -2 is both continuous and differentiable. al =

A: To Determine: fins a and b so that the function is continuous and differentiable given: we have a…

Q: Define f: Z x Z→ Z x Z by f((m, n)) = (3, 2n). Let A = {(n, 0): n ∈Z}. Fina f(A).

A: EXPLANATION : Given function  f:ℤ×ℤ→ℤ×ℤ defined by fm,n=(3,2n). given set A={(n,0):n∈ℤ}. putting the…

Q: If Tn,0(x) = ao +a¡x+ a2x² +.. he function f(x) centered at x = %3D -) True p) False

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

Q: Show that a function f : R2-R defined by f(x,y)= x3- y3 / x2+ y2, if (x,y) is not equal to (0,0)…

A: What is Continuous Function: A function is referred to be continuous at a certain point in its…

Q: Let f(x,y)=x°-4y where 0<x <1 and 0sysl. Find the minimam of value of f(x,y).

A: Answer

Q: what is the antiderivative F(x) 0f f(x)=3x^((1)/(2))-x-1that satisfies the condition F(4)=0 A,…

A: Given that:f(x)=3x1/2-x-1, F(4)=0

Q: (a) If F(v² – x²,v² – y², v² – z²) = 0 where v is a function of x, y,z show that 1 dv 1dv 1 dv 1 хдх…

A: Using chain rule.

Q: Suppose z = ƒ(x, y), where x and y are functions of t. How many dependent, intermediate, and…

A: It is observed that z=fx,y is function of x and y.

Q: Determine the linear approximation of f(x) = x* at x = 1. а. у %3D 4x - 3 O b. y=- 4 x - 3 О с. у-…

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Q: | Are fi(x) = x and f2(x) = x² orthogonal on [-2, 2] inter- val? Find c and c2 so that f3(x) =…

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Q: Let f be a differentiable function with f(0) = 1 and / f'(1) dt = x². Then Of(x) = COS X Of(x) = (x+…

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Q: (3x² -1, x 1, Determine c and d so that f(x) is continuous everywhere.

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Q: The domain of f(x) = Vx + 1 – V1 - x is O [-1, ∞) O (-∞, 1] O (-∞, –1] U [1, ∞) O [-1,1] None of the…

A: The given function is; f(x) = √(x+1) - √(1-x).

Q: Determine if f(x)=In (x² + 1) is uniformly continous on the interval I (-0, +0)

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Q: For f(x, y), find all values of x and y such that f,(x, y)= 0 and f,(x.y)= simultaneously.

A: Please refer to the image below

Q: 3. Show that there does not exist a non-constant continuous function f : R2 → R such that f(x, y) =…

A: Given:- f: R2→R

Q: Let S«) = {r²+2x}°, 121, ax+bx, x< 1. Compute for all a and b such that the function f(x) is…

A: We have to find

Q: Determine whether the functions fi(x) = cos 3r, f2(x) = x, and f3(x) = cos² r independent on the…

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Q: 4. Let F : R² → R² defined by F(x,y) = (2x +3y, –x +y). Is F a linear? Why?

A: Given

Q: 3- Prove that (y" f(x)) = "(x)y) = fy where y E Rm and f: R" → R™ %3D %3! ax ax

A: The objective is to prove that  ∂∂xyTfx=∂∂xfTxy=fxTxy where y∈ℝm  and  f:ℝn→ℝm