10. Let L : R² → R be given by 2²–y? if x + y # 0 L(x) = L(x, y) = I+y 2x otherwise Decide whether L is continuous at 0 = (0,0) E R?. Justify your claim.

Q: 1. Let h : R → R be some twice-differentiable function. Define f(x, y) = h(x) for all x ≠ 0. Show…

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Q: 12. Consider the function (z-y)2 f(x,y) = { if (x, y) = (0,0) if (x, y) = (0,0) a) Does…

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Q: Consider the function f defined by 1²+y² V12+y?+1-1 ,2 f(r, y) = %3D lim(7.y)¬(0,0) f(x, y) =

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Q: Isin , it (z, y) # (0, 0) 2. Determine if a function f : R → R defined by f (r, y) = 0, if (z, y) =…

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Q: 2. Find the degree of homogeneity of the following functions: a. f(x,y) = x³ + x³y² b.x(p,r) =…

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Q: Let f: X→ Y and g: Y→ V be functions. Show the following: (a) f is injective 3h: Y→X such that ho f…

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Q: x² - y² z = g(x, y) = x² + y² 0 for for (x, y) = (0,0) (x, y) = (0,0)

A: Given function is z=gx,y=x5−y5x4+y4 for x,y≠0,00            for x,y=0,0. We have to check whether…

Q: Indicate whether the following statement is true or false. Let f(x, y) be any c' real valued…

A: Given fx,y be the C1 real valued function defined on the rectangle R=a,b×c,d. Then we have to check…

Q: Use the definition of differentiability to show that f(x, y) = y² – xy+5x is differentiable…

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Q: → R be a function defined by 3x² + x+5 f(x) = not continuous at x = = 0.

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Q: f(x,y) be defined implicitly as a function of x and y by the 3xyz =exyz Əz Əz yay ax Əz Əz Find and…

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Q: Theorem 12. Let f be a non-zero continuous function on R such that f(x+y)-f(x) f(y) ♥x, yER. Show…

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Q: 2. Assume that h: [a, b²] → R is a continuous function and let G : [a, b] → R denot the following…

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Q: b) f(x) = x2-x %3D x²-1

A: To find out where the function is discontinous.

Q: Suppose that f: R² → R² is given by f(x, y) = (x³ - 2xy², x+y). Does f has a differentiable local…

A: Inverse mapping theorem.

Q: a sin (), it (, u) + (0, 0) x²+y? 2. Determine if a function f : R² → R defined by f (x, y) = 0, if…

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Q: If f(x,y,z) = 2x² − yz + xz², where x = 2 sint, y = t² − t + 1, and z = e−³t, find f't)

A: from chain rule of derivaties…

Q: Locate and classify the local extremum of f when f(x, y) 4.x2 – 6x – 6y² – 4y + 10xy. - - 1 1.…

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Q: 12.Find f(Vx), given the transcendental function f(x)=in x

A: The given transcendental function is: fx=lnx

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: Show that a continuous function f : [0, 1] → Z is a constant function.

A: Consider a function .We need to show that a continuous function  is a constant function.We know…

Q: Show that the function g(x, y) = is not continuous at (0,0). xy 2x² + y²¹ 0, (x, y) = (0,0) (x, y) =…

A: g(x,y)=(xy/(2x2+y2), (x, y) not equal to (0,0)               0                 ,(x, y) = (0,0)

Q: 1. Let f(u, v) be a continuously differentiable function. Let u = e*, v = e³. Show that u²fuu + v²…

A: Given that f(u,v) is a continuously differentiable function, and u=ex, v=ey then show that…

Q: 1. Is a function f(x)=√x differentiable at a) X = 1 b) X = 0 c) X = -1

A: Since you have posted multiple questions i can do first question as per our company guidelines…

Q: 5. Assume f: [0, 1] → [0, 1] is continuous with f(0) = f(1). Prove there are x, y = [0, 1] with |x -…

A: We have to assume f: [0, 1] → [0, 1] is continuous with f(0) = f(1). and prove that there are x, y ∈…

Q: 2. Let f(x,y) = {'1, = {0, y≤0ory 2x² 0<y<x¹ (a) Show that f(x,y) → 0 as (x,y) → (0,0) along any…

A: f( x, y) = 0y≤0 or y⩾x410&lt;y&lt;x4 Note that  f (0,0) = 0 . To show that f is continuous at (0,0)…