Lagrange Multipliers

1. Use Lagrange multipliers to find the extrema of the function $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=x^2+y^2$ subject to the constraint $xy=1$.
   Minima at $\left(1,1,2\right)$ and $\left(-1,-1,2\right)$.
2. Use Lagrange multipliers to find the extrema of the function $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=e^{xy}$ subject to the constraint $x^3+y^3=16$.
3. Use Lagrange multipliers to find the extrema of the function $f<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)=xyz$ subject to the constraint $x^2+2y^2+3z^2=6$.
   Maximum ${\displaystyle \frac{2}{\sqrt{3}}}$ and minimum ${\displaystyle \frac{-2}{\sqrt{3}}}$.
4. Use Lagrange multipliers to find the extrema of the function $f<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)=x^4+y^4+z^4$ subject to the constraint $x^2+y^2+z^2=1$.
5. Use Lagrange multipliers to prove that of all the rectangles with a given perimeter, the square has the largest area.
6. Use Lagrange multipliers to find the point on the plane $x-y+z=4$ that is closest to the point $\left(1,2,3\right)$.
   ${\displaystyle ⟨\frac{5}{3},\frac{4}{3},\frac{11}{3}⟩}$
7. Use Lagrange multipliers to find the points on the surface $y^2=9+xz$ that are closest to the origin.
8. Use Lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane $x+2y+3z=6$.
   ${\displaystyle \frac{4}{3}}$