= 0 for3. (True/False) If f(x, y) has a local minimum and is differentiable at (a, b), then fu(a, b)any unit vector ū.4. (True/False) Two lines in three-dimensional space either intersect or are parallel.5. (True/False) Every critical point is either a local maximum or a local minimum.6. (True/False) Two lines in two-dimensional space either intersect or are parallel.7. (True/False) For any three-dimensional vectors u and ʊ, we have |ũ × ʊ| = |ʊ × ú|.8. (True/False) Two lines in three-dimensional space parallel to a plane are parallel to one another.9. (True/False) If f(x, y) is a continuous function on a closed, but unbounded set D, then f(x, y)cannot achieve a local maximum on D.10. (True/False) For any continuous function f(x, y), we have fxy = fyx.

Since you have asked a question with multiple subparts so as per guidelines will solve the first…

Answered: 3. (True/False) If f(x, y) has a local…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose that z is an implicit function of x and y in a neighborhood of the point P = (0, −3, 1) of the surface S of equation: exz + yz + 2 = 0 An equation for the tangent line to the surface S at the point P, in the direction of the vector w = (3, −2), corresponds to:
A bug crawls on the surface z = x2 - y2 directly above a path in the xy-plane given by x = ƒ(t) and y = g(t). If ƒ(2) = 4, ƒ′(2) = -1, g(2) = -2, and g′(2) = -3, then at what rate is the bug’s elevation z changing when t = 2?
Find the parametrization of the curve defined by the intersection of the two surfacesy = x^2 − 3z^2 and x^2 + z^2 = 9, where x ≥ 0.
Find the maximum value of w=xyz on the line of intersection of the two planes x+y+z=40 and x+y-z=0.
Find a parameterisation of the curve of intersection of the surfaces: x^2 + y^4 + 2z^3 = 4 and x = y^2
The minimum value of f(x,y,z) = x^2 -2y + 2z^2 on the sphere x^2 + y^2 + z^2 =1 is ....?
You are in charge of erecting a radio telescope on a newly discovered planet. To minimize interfer-ence, you want to place it where the magnetic field of the planet is weakest. The planet is spherical, with a radius of 6 units. Based on a coordinate system whose origin is at the center of the planet, the strength of the magnetic field is given by M(x, y, z) = 6x - y2 + xz + 60. Where should you locate the radio telescope?
A particle moves along a circular path over a horizontal xy coordinate system, at constant speed. At time t1 = 4.90 s, it is at point (4.60 m, 5.00 m) with velocity (2.00 m/s)ĵ and acceleration in the positive x direction. At time t2 = 13.6 s, it has velocity (–2.00 m/s)î and acceleration in the positive y direction. What are the x and y coordinates of the center of the circular path? Assume at both times that the particle is on the same orbit.
find the points of the surface x^2-y^2+z^2=1 which corresponds to the extreme value of f(x,y,z)=x-y-z.
The discriminant ƒxx ƒyy - ƒxy 2 is zero at the origin for each of the following functions, so the Second Derivative Test fails there. Determine whether the function has a maximum, a minimum, or neither at the origin by imagining what the surface z = ƒ(x, y) looks like. Describe your reasoning in each case. a. ƒ(x, y) = x2y2 b. ƒ(x, y) = xy2 c. ƒ(x, y) = x3y3 d. ƒ(x, y) = 1 - x2y2 e. ƒ(x, y) = x3y2 f. ƒ(x, y) =x4y4
Consider the curve y = 10 + 4x − x^(2) at (x, y) = (3, 13). Find a vector, v, that has length 4 and is parallel to the tangent line to y = 10 + 4x−x^(2) at x = 3.
Find the minimum distance from the point (4, 0, 6) to the plane x − y + z = 3.

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3. (True/False) If f(x, y) has a local minimum and is differentiable at (a, b), then fu(a, b) any unit vector u. = 0 for 4. (True/False) Two lines in three-dimensional space either intersect or are parallel. 5. (True/False) Every critical point is either a local maximum or a local minimum.