6.8E:

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Exercise $\PageIndex{1}$

For the following exercises, use the method of Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints.

1) $f(x,y)=x^2y ;x^2+2y=6$

2) $f(x,y,z)=xyz; x^2+2y^2+3z^2=6$

Answer: maximum: $\frac{2}{\sqrt{3}$, minimum: $-\frac{2}{\sqrt{3}$

3) $f(x,y)=xy;4x^2+8y^2=16$

4) $f(x,y)=4x^3+y^2;2x^2+y^2=1$

Answer: maximum: $√2$, minimum: $ -sqrt{2}$

5) $f(x,y,z)=x^2+y^2+z^2,x^4+y^4+z^4=1$

6) $f(x,y,z)=yz+xy,xy=1,y^2+z^2=1$

Answer: maximum: $32$, minimum = $11$

7) $f(x,y)=x^2+y^2,(x−1)^2+4y^2=4$

8) $f(x,y)=4xy,x^{2}+y^{2}=1$

Answer: maximum: $2$, minimum = $-2$

9) $f(x,y,z)=x+y+z,x+y+z=1$

10) $ f(x,y,z)=x+3y−z,x^2+y^2+z^2=4$

11) $f(x,y,z)=x^2+y^2+z^2,xyz=4$

Exercise $\PageIndex{2}$

1) Minimize $f(x,y)=x^2+y^2$ on the hyperbola $xy=1.$

Answer: $2$

2) Minimize $f(x,y)=xy$ on the ellipse $b^2x^2+a^2y^2=a^2b^2.$

3) Maximize $f(x,y,z)=2x+3y+5z$ on the sphere $x^2+y^2+z^2=19$

Answer: $19 \sqrt{2}$

4) Maximize $f(x,y)=x^2−y^2;x>0,y>0;g(x,y)=y−x^2=0.$

5) The curve $x^3−y^3=1$ is asymptotic to the line $y=x.$ Find the point(s) on the curve $x^3−y^3=1$ farthest from the line $y=x.$

Answer: (12√3,−12√3)

6) Maximize $U(x,y)=8x^4/5y ;4x+2y=12$

7) Minimize $f(x,y)=x^2+y^2,x+2y−5=0.$

Answer: $f(1,2)=5$

8) Maximize $f(x,y)=6−x^2−y^2; x+y−2=0.$

9) Minimize $f(x,y,z)=x^2+y^2+z^2;x+y+z=1.$

Answer: 13

10) Minimize $f(x,y)=x^2−y^2$ subject to the constraint $x−2y+6=0.$

11) Minimize $f(x,y,z)=x^2+y^2+z^2$ when$ x+y+z=9$ and $x+2y+3z=20.$

Answer: minimum: f(2,3,4)=29

Exercise $\PageIndex{3}$

use the method of Lagrange multipliers to solve the following applied problems.

1) A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the diagram. If the perimeter of the pentagon is 1010 in., find the lengths of the sides of the pentagon that will maximize the area of the pentagon.

A rectangle with an isosceles triangle on top. The side of the isosceles triangle with the two equal angles of size θ overlaps the top length of the rectangle.

2) A rectangular box without a top (a topless box) is to be made from 1212 ft² of cardboard. Find the maximum volume of such a box.

Answer

The maximum volume is 44 ft³. The dimensions are 1×2×21×2×2 ft.

3) Find the minimum and maximum distances between the ellipse $x^2+xy+2y^2=1$ and the origin.

4) Find the point on the surface $x^2−2xy+y^2−x+y=0$ closest to the point $(1,2,−3).$

Answer: (1,12,−3)

5) Show that, of all the triangles inscribed in a circle of radius $R$ (see diagram), the equilateral triangle has the largest perimeter.

A circle with an equilateral triangle drawn inside of it such that each vertex of the triangle touches the circle.

6) Find the minimum distance from point $(0,1)$ to the parabola $x^2=4y.$

Answer: 1.0

7) Find the minimum distance from the parabola $y=x^2$ to point $(0,3)$.

8) Find the minimum distance from the plane $x+y+z=1$ to point $(2,1,1).$

Answer: 3–√3

9) A large container in the shape of a rectangular solid must have a volume of 480480 m³. The bottom of the container costs $5/m² to construct whereas the top and sides cost $3/m² to construct. Use Lagrange multipliers to find the dimensions of the container of this size that has the minimum cost.

10) Find the point on the line $y=2x+3$ that is closest to point (4,2).

Answer: (25,195)

110 Find the point on the plane $4x+3y+z=2$ that is closest to the point (1,−1,1).

12) Find the maximum value of $f(x,y)=sinxsiny,$ where $x$ and $y$ denote the acute angles of a right triangle. Draw the contours of the function using a CAS.

Answer: 12

13) A rectangular solid is contained within a tetrahedron with vertices at (1,0,0),(0,1,0),(0,0,1),(1,0,0),(0,1,0),(0,0,1), and the origin. The base of the box has dimensions $x,y$ and the height of the box is $z.$ If the sum of $x,y$, and $z$ is 1.0, find the dimensions that maximize the volume of the rectangular solid.

14) [T] By investing x units of labour and y units of capital, a watch manufacturer can produce $P(x,y)=50x0.4y$ watches. Find the maximum number of watches that can be produced on a budget of $$20,000$ if labour costs $100/unit and capital costs $200/unit. Use a CAS to sketch a contour plot of the function.

Answer

Roughly 3365 watches at the critical point (80,60)(80,60)

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Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3}\)