Suppose at x = x, Vf(x) = 0. What can you say about x if (a) f(x) is convex? (b) f(x) is concave? (c) V²f(x) is indefinite? (d) V²f(x) is positive semidefinite? (e) V²f(x) is negative semidefinite?

Q: How would I solve ∭xzdV, where E is bounded by the planes z = 0, z=y, and the cylinder x2 + y2 = 1…

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Q: Is the point a = [0, 1]T a local minimizer? %3D

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Q: 6. Evaluate (7xi - zk) · ndA over the sphere S : x² + y² + z² = 4 (u) by the Divergence Theorem, (b)…

A: The divergence theorem states that  ∬SF·ndA=∭V∇·FdV Where, F is the vector field, V is the given…

Q: Use Lagrange multipliers to find the maximum value of f(x, y, z)= xyz where x >0, y>0 and z>0,…

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Q: Use Taylor's formula for f(x,y) at the origin to find quadratic and cubic approximations of f(x,y) =…

A: The function fx,y=11-3x-y Then the taylor's formula for fx,y is, fxx,y=31-3x-y2fyx,y=11-3x-y2 Then…

Q: 1. Find where f(x, y) = 4x² + 8y? – 4x – 8y +5 has absolute extreme values over the square with…

A: To calculate the extreme values of f(x,y)bounded by the square, the following calculations need to…

Q: find the extreme values (absolute and local) of thefunction over its natural domain, and where they…

A: Given that,

Q: (A) Suppose C= 4x²y + i3x³y? . Find divergence of C.

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Q: 3. Use the method of Lagrange multipliers to find the minimum distance between the paraboloid z = x²…

A: (3)Let (x,y,z) lies on parabolaThe distance d=x-32+y2+z2such that z=x2+4y2Lagrabge multiplier…

Q: -JIL e √x² + y² dv, where E is the region that lies inside the cylinder x² + y² = 4 and between the…

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Q: 3. Use the method of Lagrange multipliers to find the minimum distance between the paraboloid z = x?…

A: Given,                               z=x2+4y2and the point (3, 0, 0).

Q: Given etye -Z+ xy - e-0, use e =0, use impicit partial denvatve to finid 2ž at the Point (1 2t

A: Explanation of the solution is given below....

Q: 2. Use the method of Lagrange multipliers to show that for x, y ≥ 0, we have z"+"> (²+)" 2

A: Introduction: For minimizing a multi-variable function under some certain constraint or constraints,…

Q: 2. Let f: R? → R be the function which satisfies for all (x, y) € R² that f (x, y) = xy. Use the…

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Q: Consider a circle of radius 2 centered at the origin. Use the Lagrange multiplier method to find the…

A: It is given that circle is of radius 2 and centered at origin. The function isfx,y=3x-y+5.

Q: Show that y^3dx+3xy^2dy=0 is separable and exact

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Q: 3. Find the family of oblique trajectories that intersect the family of the paraboles y =cx at angle…

A: given family of parabolas y2=cx at angle θ=60° claim- find the family of oblique trajectories that…

Q: Test the following surface for maxima minima and saddle point Z= 5+xy+x²+2y+3x

A: First we will find the gradient of the given function using the gradient we will find the critical…

Q: Find the absolute maximum and minimum of f(x,y)= 4xy^2 - (x^2)(y^2) - xy^3 on the closed triangular…

A: f(x,y)=4xy2-x2y2-xy3fx=4y2-2xy2-y3fx=04y2-2xy2-y3=04-2x-y=0…

Q: Test the function def ined by f (x,y) = - x- 4y for its homogeneity. If it is homogeneous, give its…

A: Find your answer below

Q: 4. Find the orthogonal trajectories of the family of parabola described by the equation (y– 3) = Cx.…

A: The solution of the problem is given in the next step.

Q: The orthogonal trajectories of the family of curves y= cx² is A x=ky? B 2x2+y2 = k © x2 = ky? x2…

A: The given family of curve y=cx2

Q: (ii) Find the extremum of Z= x2 +y2 subject to x + 4y 2 and examine whether the %3D %3D stationary…

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Q: find the maximum value that f(x, y, z) = x^2 + 2y - z^2 can have on the line of intersection of the…

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Q: Let C be the boundary curve counterclockwise of the semidisk x² + y? < 9, y 2 0. Then f.(-2ydx +…

A: A detailed solution is given below.

Q: Suppose that a thin metal wire in the shape of a semicircle occupies the curve C, where C is the top…

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Q: . Use the method of Lagrange multipliers to show that for x, y > 0, we have a" +y" > (팔1)" 2

A: We will first try to minimise (xn + yn)/2 using Lagrange multipliers then show that its minimum…

Q: How would I solve ∭xzdV, where E is bounded by the planes z = 0, z=y, and the cylinder x2 + y2 = 1…

A: The given equation of the cylinder us x2+y2=1 in the half space y≥0. Calculate the value of ∫∫∫xzdV,…

Q: 7) Let Z.= 3x2 - y² . Find all points at which |VZ| = 6 %3D

A: There is an ellipse,  all points on the ellipse are the solutions of given equation.

Q: 2) Use Lagrange multipliers to find all extreme values of f (x, y,z)= 2x+2y+z subject to x + y² +z²…

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Q: 2. Find the orthogonal trajectories of the family of parabolas y = cx O In(x² + y² ) = c O 2r? + y²…

A: Explanation of the answer is as follows

Q: Sketch the solid described by the given inequalities. 0srs 8, -T/2 se S T/2, 0szs 1. 1.0 1.0 z05…

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Q: a) Find the extreme values of f(x, y) = x' + y' - 3xy +1 by using the Second Derivatives Test. b)…

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Q: Find the surfaces on which the function u = 2 (x2 + y²)/z is constar

A: Since u is a constant so, x2+y2z=4a , where a is an arbitrary constant.

Q: For x,y,z > 0, use the Lagrange Multiplier Method to find the minimum value of 1 1 1 + + x y Subject…

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Q: 4. (a) Use the trapezoidal rule to numerically integrate from a = 0 to b= 2 /(z)-0.2+25 +3r

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Q: Find x = that minimizes q(x) = x² – xy + y² subject to %3D the condition ||x|| = 1 , %3D

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Q: maximize f(x,y)=xy subject to the contraint 6x+y=18  the last question is what is the maximum valye…

A: max . fx,y=xyS.t 6x+y = 18     6x+y=18⇒ 6x+y-18=0∴ gx,y=6x+y-18=0    Fx,y,L=fx,y-L g x,y⇒…

Q: The orthogonal trajectories of y = ce5" where c is arbitrary constant. is - 2x + 2C = 5y? 2x + 2C =…

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Q: 4) Find the orthogonal trajectories of the family of parabolas y – 3 = c(x + 1)².

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Q: Find the maximum of f(x) = 4x – 1.8x2 + 1.2x³ – 0.3x4 using Newton's method %3D (xo = 3, ɛs = 1%).

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Q: For functions f(x,y,z) = 5xyz and g(x,y,z) = x2 + 3y? + 4z2 – 7. write the Lagrange multiplier…

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Q: 2 : Use Lagrange multipliers method to show that the maximum value of f(x, y, z) = x²y²z² on the…

A: In lagrange method, the Critical points are the points that cancel the first order partial…

Q: Let C be the boundary curve counterclockwise of the semidisk x² + y? 0. Then § (-2ydx + 3xdy) = a)…

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Q: Verify Young's Theorem z = In(2x + y) y? x2 +e(x+2y) y

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Q: The orthogonal trajectories of the family of curves y= cx² is A x² = ky? x² +2y² =k В © x=ky? D 2x2+…

A: In this question we have to find the orthogonal trajectories of the family of the curves y=cx2

Q: Show that the subtangent to y = x' at x = a is equal to a/3.

A: Definition used -                            Length of subtangent =ydxdy

Q: 1. Find the linearization of r – z at a = 2.

A: Find the linearization of x3-x at a=2.

Q: state where in the ty-plane the hypotheses of Theorem are satisfied 6. y′=ln|ty|1−t2+y2

A: A hyperbola is an open curve with two branches, which are symmetric and are mirror images of each…