Let f(x,y)=1+x2−cos(5y). Find all critical points and classify them as local maxima, local minima, saddle points, or none of these.
5. Consider the function f: R -> R given by f (x, y) := e°+v* _ 4. (a) Sketch the level curves of f. (5 marks) (b) Find Vf, the gradient of f, and determine at which points Vf is zero. Remark: These points are called the critical points of f (5 marks) (c) Determine whether the critical points of f are local minima, local maxima, or saddle points by considering the level curves of f. (5 marks) (d) Calculate...
Let f(x, y) = x(x – 1) + y2. (a) [1 point] Sketch the level curves of f. (b) [2 points] Compute the gradient of f, and sketch it as a vector field. (c) [3 points) Find all critical values of f and classify them as local maxima, local minima, or saddle points.
2. For each function, find all critical points and use the Hessian to determine whether they are local maxima, minima, or saddle points. (a) f(x,y,z) = x — 2 sin x – 3yz (b) g(x, y, z) = cosh x + 4yz – 2y2 – 24 (c) u(x, y, z) = (x – z)4 – x2 + y2 + 6x2 – 22
only for part e A) Unconstrained optimization: 1) Find the local maxima, local minima and saddle points of the following functions: a)f(x, y)=x²+ y2+2x–6 y+6 b)f(x,y)=(x-1)2-(y-3)? c)f(x,y)=x2-y2–2x-4 y-4 d)f(x,y)=2xy-5x²-2y +4x+4y-4 e)f(x,y)=e(x²+y?)
3. (28 points) Let f(x,y) = 2x3 - 6xy+3y- be a function defined on xy-plane. (a) (6 pnts) Find first and second partial derivatives of f. (b) (10 pnts ) Determine the local extreme points of f (max., min., saddle points) if there is any. (C) (12 pnts) Find the maximum and minimum values of f over the closed region bounded by the lines y = -x, y = 1 and y=r
2. Let f(x,y) = 2x2 - 6xy + 3y2 be a function defined on xy-plane (a) Find first and second partial derivatives of (b) Determine the local extreme points off (max., min., saddle points) if there are any. (c) Find the absolute max. and absolute min. values of f over the closed region bounded by the lines x= 1, y = 0, and y = x
u=2i-j+k v = 37 - 4k w = -51 +7 QUESTION 1)Find the volume of the parallel face determined by the vectors QUESTION 2) f(x, y, z) = xy + y2 + zx a) Find the gradient vector of function f b) Calculate the gradient vector at point P (1, -1, 2) of function f. c) Direction in the direction of the vector v = 3i + 6j - 2k at point P (1,-1,2) of the function f find the...
will upvote if answers are correct with working Question 24 4 pts Find an equation of the tangent plane to the surface x2 + y2 += 704 at the point (8. - 8, 16). 16(x-3)-96y+8)+ 32-16) = 0 16(x-8)- 48(y+8)+ 32(2-16)=0 16x-8)+ 963 +8)+ 32(-16)0 16x-8)-96(+8)+ 163-16) = 0 16x-8)-96y+8)- 327-16) = 0 Question 25 4 pts Examine the function f(x,y)=x) - 6xy + y + 3 for relative extrema and saddle points. saddle points: (0,0,3),(2,2,-5) relative minimum: (0, 0,...
Problem 2. Let be the quarter torus with outward normal. Use the parameterization r(u, v) = (4 + 2 cos(v)) cos(u)i + (4 + 2 cos(u)) sin(u)j + 2 sin(v)k, for 0 Susand 0 <0527 (a) Find a parameterization for each of the curves forming the boundary of E. Make sure the orientation of the curves match the orientation induced by S. (b) Let F(x, y, z) = xyi+yzj+rzk. Evaluate S/.( VF) ds.