Question

We will find the maximum and minimum values of f(x, y) = xy on the ellipse (x^2)/4+ y^2 = 1 (so our constraint function is g(x, y) = (x^2)/4 + y^2 ).

a) Find ∇f and ∇g.

b) You should notice that there are no places on the ellipse where ∇g = 0, so we just need to find the points on the ellipse where ∇f(x, y) = λ∇g(x, y) for some λ. Find the points on the ellipse where fx(x, y) = λgx(x, y) and fy(x, y) = λgy(x, y) for some λ. Often a good strategy to use is to solve for λ in one of the components, and then plug this in for λ in the other component (this eliminates the λ’s and allows us to solve for just x and y). Don’t forget that we want the x and y points where (x^2) /4 + y^2 = 1.

c) Plug the points from the previous step into f(x, y) to find the maximum and minimum (you should have four total points, two of these will be the maximum and two will be the minimum).

Answer #1

Find the maximum and minimum values of f(x,y)=xy on the ellipse
2x2+y2=4.

Find the maximum and minimum values of f(x,y)= xy on the ellipse
9x^2+y^2 = 8.

Use the Lagrange Multipliers method to find the maximum and
minimum values of f(x,y) = xy + xz subject to the constraint x2 +y2
+ z2 = 4.

Use Lagrange multipliers to find the maximum and minimum values
of
f(x,y)=xy
subject to the constraint 25x^2+y^2=200
if such values exist.
Enter the exact answers. Which is global maximum/global minimum?
Enter NA in the appropriate answer area if these do not apply.

Use Lagrange multipliers to find the maximum and minimum values
of f(x,y)=6x+y on the ellipse x2+16y2=1

The graph of the tilted ellipse is x^2 - xy + y^2 - x - y =
2
a) Find the points on the ellipse where the tangent line is
horizontal
b) Find the points on the ellipse where the tangent line is
vertical
c) What are the dimensions of the box containing the
ellipse?
d) What are the coordinates of the four corners of the box
containing the ellipse?

Use the method of Lagrange multipliers to find the maximum and
minimum values of F(x,y,z) = 5x+3y+4z, subject to the constraint
G(x,y,z) = x2+y2+z2 = 25. Note the
constraint is a sphere of radius 5, while the level surfaces for F
are planes. Sketch a graph showing the solution to this problem
occurs where two of these planes are tangent to the sphere.

find the absolute maximum value and absolute minimum values of
the function f(x,y)4xy^2-x^2y^2-xy^3 on the set D, where D is the
closed trianglar region in the xy-plane with certices
(0,0)(0,6)(6,)0

Find the absolute maximum, and minimum values of the function:
f(x, y) = x + y − xy Defined over the closed rectangular region D
with vertices (0,0), (4,0), (4,2), and (0,2)

Find the absolute maximum and minimum values of
f(x,y)=2x^2+y^2-xy^2 on the triangular region shown with vertices
(0,0), (0,4) and (4,4).

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