4.8 Lagrange multipliers

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

$f\left(x,y,z\right)=xyz,x^2+2y^2+3z^2=6$

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

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

$f\left(x,y,z\right)=x^2+y^2+z^2,x^4+y^4+z^4=1$

$f\left(x,y,z\right)=yz+xy,xy=1,y^2+z^2=1$

$f\left(x,y\right)=x^2+y^2,{\left(x-1\right)}^2+4y^2=4$

$f\left(x,y\right)=4xy,\frac{x^2}{9}+\frac{y^2}{16}=1$

$f\left(x,y,z\right)=x+y+z,\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$

$f\left(x,y,z\right)=x+3y-z,x^2+y^2+z^2=4$

$f\left(x,y,z\right)=x^2+y^2+z^2,xyz=4$

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

Minimize $f\left(x,y\right)=xy$ on the ellipse $b^2{x}^2+a^2{y}^2=a^2{b}^2.$

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

Maximize $\begin{array}{c}f\left(x,y\right)=x^2-y^2;x>0,y>0;\hfill \\ g\left(x,y\right)=y-x^2=0\hfill \end{array}$

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 $y=x.$

Maximize $U\left(x,y\right)=8x^{4\text{/}5}{y}^{1\text{/}5};4x+2y=12$

Minimize $f\left(x,y\right)=x^2+y^2,x+2y-5=0.$

Maximize $f\left(x,y\right)=\sqrt{6-x^2-y^2},x+y-2=0.$

Minimize $f\left(x,y,z\right)=x^2+y^2+z^2,x+y+z=1.$

Minimize $f\left(x,y\right)=x^2-y^2$ subject to the constraint $x-2y+6=0.$

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

A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the diagram. If the perimeter of the pentagon is $10$ in., find the lengths of the sides of the pentagon that will maximize the area of the pentagon.

A rectangular box without a top (a topless box) is to be made from $12$ ft ² of cardboard. Find the maximum volume of such a box.

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

Find the point on the surface $x^2-2xy+y^2-x+y=0$ closest to the point $\left(1,2,\mathrm{-3}\right).$

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

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

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

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

A large container in the shape of a rectangular solid must have a volume of $480$ m ³ . The bottom of the container costs $5/m ² to construct whereas the top and sides cost $3/m ² to construct. Use Lagrange multipliers to find the dimensions of the container of this size that has the minimum cost.