3. Show that there does not exist a non-constant continuous function f : R2 → R such that f(x, y) = 5 for all (x, y) E R² with a? + y² < 1.

Q: Find an upper bound for ||L|| where L: R² → R³ with L(x, y) = (2x + y,x + 3y, 3x – 2y) -

A: Here we use the definition of boundedness of a linear operator.

Q: Suppose that F(x, y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that…

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Q: -Find the D.f ofr (x, y) = (x² – y)' at Po(3, 1) with

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Q: Let fn(x) = E s;(x) Y; be a linear smoother. Find a condition on the weights s;(x) so that fn(x) is…

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Q: 9. Use the chain rule for multivariable functions to find where f(u, v) = u + v u(x, v, z) = x+y,…

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Q: What is the complete domain D and range R of the following multivariable functions: a. w(x, y) = Vy…

A: a. we have  W(x,y)=y-4x2 we have to find the value of (x,y)  where W(x,y) is defined  y-4x2  is only…

Q: Show that the function f (x, y)=(2 + x-y) / [1+ 2x ^ 2)+(3y ^ 2)] ∈R has a limit at (0,0).

A: I have provided a solution in step2.

Q: 2- Obtain the function x to minimize the following functional. = [ c«f + *)dt J = St: x1 = -X2 ,…

A: J=∫01x12+x22dt-------(1) x1˙=-x2-----(2) x(0)=c, x(1)=d here c,d are fixed number

Q: If f: (a,b]-R has a local minimum at pe (a,b) and f is differentiable at p, then prove that f'(p) =…

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Q: 4. If f is a differentiable function of x and y with w = f(x, y), x = r cos 0 and y = r sin 0, show…

A: Given: w=fx,y, x=rcosθ, y=rsinθ If w is a multi variable function w(x,y,a,........z) thus the…

Q: Let f(x,y)=3x2-2xy+y2-8y  Find the absolute extrema over [0,1]x[0,2]

A: Consider the given function fx,y=3x2-2xy+y2-8y First, find the partial derivatives of the function…

Q: Find the local linear approximation L (x)of the function f (x)= x² +3 at x = 1

A: Given that: f(x)=x2+3

Q: Suppose that f(x, y) : at which {(x, y) | 0 < x < 4, – x < ys va}. 1+ x D

A: Consider a given function:  The objective is to evaluate the following double integral of fx, y…

Q: 8. What is the relationship between a and b so that F(x, y) = (ary, 3ax + bry) is source free?

A: That's easy. Have a great day!!!

Q: x+y if (x, y) # (0,0) (1 if (x,y) (0,0) Q.3 ( 3+3+4) (a) Prove that the function f(x, y) is not…

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Q: f(z) =|z| is continuous in C 0 صواب ihi O

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Q: * (0,0) is not continuous if (x, y) Q.3 ( 3+3+4) (a) Prove that the function f(x,y) = {2x+y (i if…

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Q: Let f(x,y)=3x2-2xy+y2-8y  Find the absolute extrema over [0,1]x[0,2].

A: We have f(x, y)=3x2-2xy+y2-8y find critical points ∂f∂x=6x-2y=0 ... (1)∂f∂y=-2x+2y-8=0  ...(2)solve…

Q: At what points (x, y, z) in space are the functions continuous?  a. ƒ(x, y, z) = x2 + y2 - 2z2 b.…

A: Continuity: A function f is continuous at the point (a, b, c) if,                            For…

Q: 5. Let F(x, y) = x² + y² + xy + 3. Find the absolute maximum and minimum values of F on D = {(x, y)|…

A: The main objective is to find absolute maximum and minimum of f(x,y) subject to given condition.

Q: ( x+y if (x,y) # (0,0) Q.3 ( 3+3+4) (a) Prove that the function f(x,y) = 2x+y is not continuous (1…

A: We apply two path test for proving the discontinuity of function

Q: At what points (x,y,z) in space are the functions continuous? a. h(x,y.z) = In (3z° - 4x- 5y- 4) 1…

A: Please rate

Q: Suppose f(x) is continuous on R such that F(x) is an antiderivative of f(x). Furthermore i Find F(1)…

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Q: For the IVP, y'=f(x,y), y=yo, x=xo, iff and df/dy are continuous discontinous defined undefined then…

A: Here we have to choose the correct option.

Q: x²y- ry f(x, y) is continuous at (1, 1). x- y+1

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Q: d dx y' z' y" z"

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Q: 2. Show that the function g defined by VVT y² + √² 2, is not continuous at (0,0). (z,y) = (0,0)…

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Q: If z is a function of x and y implicitly defined by əz əz find and əx ду at (x, y, z) = (0, 2, -1).…

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Q: Let f (x,y, z) = xyz2 + 1 and suppose that x(t) = 2 – t, y(t) = In(t), and z(t) = 3t. df Use the…

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Q: Let f(x, y, z) = ln(x² + z) + y²e#z – cos(2yz). Find fzya.

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Q: fhow thot te function f depined by f(XrY) = 1 (x,Y) =(1,-リ - 1) %3D x2 +y (x, y) F (I,-1) * t y is…

A: We use the sequential definition of continuity to solve this problem.

Q: Find dy any ay of Ay of y =f(x) = x² +x at x= 10 when ax =0.1.

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Q: 2. Find whether the function F-(* - y' )i + (v² - 3x)j + (:² - xy is irrotational.

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Q: Let f(x, y) = xcosy- y'e*y. fxy at (1,0)is equal to: %3D -2/3 -2 1 0.

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Q: A necessary condition for the functional defined by X2 = dx f(x, Y(x),Y'(x)) to have a local minimum…

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Q: (a) Let Ty if +y +0, g(x, y) = 0 if r = y = 0. Prove that g(r, y) is continuous at (0,0).

A: To Prove: g(x,y) is continuous at (0,0)

Q: 10.Show that F(x, y) = (1 + xy)e*yi + (ev + x?e*y)j is conservative and find a function "f" such…

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Q: Let F(x,y,z) = z* + x+ y sin x + x In z 0, where z fx.y), then at (0,0,1) and are dy equal to: ax

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Q: Suppose that z is a differentiable function of x and y with dz (1, 1) = 162 and(1, 1) = 6. dz ду dx…

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Q: 3. Prove by using the Squeeze Theorem, or otherwise: 1 t2 dt is continuous at 0. T sin t (a) (b) 1+…

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: Show that f(z) = x³+i(1−y)³ is differentiable only at x = 0, y = 1 and f'(z) = 0.

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Q: J. Suppose that X and Y have a continuous joint distribu- tion for which the joint p.d.f. is as…

A: a)Here,fxX=∫01fx,ydy=∫01x+ydy=∫01xdy+∫01ydy=xy01+y2201=x+12…

Q: ) Consider the function f(r) = f(1) has a critical number A %3D f"(A) = Thus we conclude that f(x)…

A: Critical point f(x) is the number at which first derivative is zero.…

Q: Find fy for f(x, y) = (In x) (sin x² y). O fy = (In x)(cos x²y)(2xy) O fy = (cos x2y)(x²) O fy = (In…

A:

Q: B// If w= f(x,v,z) is differentiable and x,y and z are differentiable functions of t, the w is a…

A: w=f(x,y,z) is a differentiable and x,y, and z are differntiable function of t.…

Q: 2. i) Give an example of a nonzero continuous function f on [0,1] such that f(x)dx = 0.

A: We have to give an example of a nonzero continuous function f  on [0,1] such that  ∫01f(x)dx = 0

Q: 1. Show that the function x2 – y2 2xy f(x,y) ={ x2 + y? if (x,y) # 0 if (x, y) = 0 is continuous at…

A: If for every ε&gt;0, there exists a δ&gt;0, such that if |x-a|&lt;δ, |y-a|&lt;δ, then the function…

Q: Let Ly = y" + py' + qy. Suppose that yi and y2 are two functions such that Lyi = f(x) and Ly2 =…

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Q: Let f(x,y, z) = In(x²2+ z) + y²e*z – cos(2yz). Find fzyx·

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Q: 4. Suppose that c is a constant and that u is a continuous, differentiable function of x. Show:…

A: prove of the above question is given below...