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- Find the maximum off (x, y, z) = xyz subject to the constraint g(x, y, z) = 2x + y + 4z = l.graph the points z = x + iy that satisfy the givenconditions.a. | z | = 2 b. | z | < 2 c. | z | > 2Find the extreme values of f(x, y, z) = x + 2y + 3z subject to the two constraints x + y + z = 1 and x 2 + y2 + z2 = 1.
- Find the minimum value of h(x, y) = (x − 2)2 + 2xy − (y + 4)2 constraint y=x. subject to theFind the extreme values of f(x,y,z)=yz+xy+1 subject to both constraints xy=1 and y^2+z^2=18.minimize f(z,y,x) =2x2+5y2+6z2 subject to the constraint 4x+5y+8z=28. when the minimum occurs, what is a. x? b. y? c.z ? what is the minimum value of f(x,y,z)?
- The function f(x, y, z) = x2 + y + z has an absolute maximum and absolute minimum subject to the constraint 2x2 + 2y2 + z2 = 2. Use Lagrange multipliers to find these values.Find the extreme values of the function f(x,y) subject to the given constraint. f(x, y) = 4x + 6y, x2 + y2 = 13 A) Maximum 36 in (3, 4); minimum 0 at (0, 0)B) Maximum 26 in (2, 3); minimum 0 at (0, 0)C) Maximum 36 in (3, 4); minimum -36 in (-3, -4)D) Maximum 26 in (2, 3); minimum -26 in (-2, -3)Find the values of x, y, z that minmize x2+y2+z2-2x-2y-2z+3 subject to the constraint 38−x−3y−z=0. The given function subject to the constraint 38−x−3y−z=0 has a minimum at x=___________, y=__________, and z=____________.
- 22.Prove that for the system dxdt=F(x,y),dydt=G(x,y) there is at most one trajectory passing through a given point (x0, y0). Hint: Let C0 be the trajectory generated by the solution x = ϕ0(t), y = ψ0(t), with ϕ0(t0) = x0, ψ0(t0) = y0, and let C1 be the trajectory generated by the solution x = ϕ1(t), y = ψ1(t), with ϕ1(t1) = x0, ψ1(t1) = y0. Use the fact that the system is autonomous, and also the existence and uniqueness theorem, to show that C0 and C1 are the same.Find the maximum and minimum values of z = 16x + 34y, subject to the following constraints. 2x + 10y ≤ 80 6x + 2y ≤ 72 3x + 2y ≥ 6 x ≥ 0, y ≥ 0 A) The maximum value is z = ___________ , at (x, y) = (_,_) B) The minimun value is z = ___________ , at (x, y) = (_,_)Use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. f(x,y,z)=x^2+y^2+z^2, xyz=4