3: Let u(x, y) and v(x, y) be two twice differentiable functions such that ux = Vy, Uy Show that u(x, y) has also a saddle point at P. Hint: First, show that P is a critical point of u(x, y) and then apply 2nd derivatiyę test. = -Vg and v(x, y) has a saddle point at some P.

Q: fany)=y+x3-6xー6y 2. y+2 function is given a) Fird, your critical Points of f ? b)Find all the…

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Q: 3. f(x, y) = 5x¹ − 4x5 + 5y¹. Ans.: local minimum 0 at (0, 0), saddle at (1,0)

A: 3. Given that the function                          f(x, y)=5x4-4x5+5y4.

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Q: Let C be a portion of a curve from (2,0) to (2,2). Let F(r,y) = (tan(x²) + e²v¸ 2re²v

A: Introduction: A vector field is conservative if ∂P∂y=∂Q∂x, Where f(x,y)=Pi+Qj.

Q: True/False: The function f(r, y) = xy + + 4 has a saddle point at (2,1). O True O False

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Q: Function g(x,y) is differentiable everywhere. Classify its critical point (0,-2) if the following is…

A: Given that g(0.-2)=-2gxx(0,-2)=-1gyx(0,-2)=2gyy(0,-2)=-5

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Q: 2. Let closed curve C consist of the line segment from (0,0) to (2, 1), followed by the line segment…

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Q: 2. Find normal line to zero level surface of: f(x,y, z) = exy, Po(2,5) %3D

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Q: 4. Find the locations (x, y) AND function values f(x, y) for all relative extrema and saddle points…

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Q: b) Given z = f(x -y, 2x - 2y, 3x – 3y). Show that dz dz 0. ду

A: The solution is given as

Q: True/False: The saddle point of the of the function f(x,y) = x² – e occurs at (0,0). - O True O…

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Q: af of H.W: Find and for the functions: b əx ду 1. f(x, y) = (xy-1)² 3. f(x, y) = (x²-1)(y + 2)…

A: Given functions are 1. f(x, y)=(xy-1)2 3. f(x, y)=(x2-1)(y+2) 4. f(x, y)=x(x2+y2)   To Find: ∂f∂x…

Q: Find the extrema and/or the saddle points of the functions a) f(x, y) = ex cos y b) f(x, y) = x² +…

A: Note: As per the honor code, the first question can be answered since the exact one wasn’t…

Q: Find the extrema of the function f (x, y) =x² + y² + x²y + 4 in the domain D={(X, Y), 1X1 <1, iyi <…

A: See the attachment

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Q: A) Find and classify the critical points of f(z, y) = x³ + y3 – 6y? – 3.x + 9 using the first and…

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

Q: utt - (rux + X(a,b) Utx)x - Byt = 0, x €]0, 1[, t > 0, Ytt - (Yx + X(a,b) Ytx)x+ But = 0, x €]0, 1[,…

A: As per the question, we consider a system governed by a set of partial differential equations with…

Q: Find the local linear approximation to f(x, y) = 4x² + y² at P(2,3)

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Q: The function f whose gradient vector is vf(x,y) = (xIn(v + 7), 7x + y + 7> has two critical points…

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Q: 2. Suppose that X and Y have the following joint PMF: P» (x,y) Y X 1 2 0.10 4 0.15 6 0.10 0 0.15…

A: The joint PMF of X and Y is  PX,Y(x,y)              Y X 0 1 2 2 0.15 0.10 0.10 4 0.20 0.15…

Q: The gradient of f(x, y) = ² * sin(2y) at (x, y) = (2, 1) is defined as followed: Vf(x, y) = (fz(2,…

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Q: Let f(x, y, z) = xyz Find the direction in which the function increase most rapidly at the point…

A: Given: fx,y,z=xy2z3 The gradient of a function if given by ∇f(x,y,z) =∂f∂x,∂f∂y,∂f∂z.

Q: f (x, y) = xª – 2.x² + y² – 4y + 5. a) Use the second derivative test to find the critical points b)…

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Q: 6) Evaluate ff(x,y) dx, ff(x,y) dy and ff(x,y) ds on the indicated curve C. f(x, y) = 3x² + 6y²; y =…

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Q: Let f a function that allows second derivatives: Vf(x, y) = (a²x-a²x², y² + ay) With a <0, it can be…

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