Example 4.9.1

Find the extreme values of $f\left(x\,y\right)\=2 x^2\+3 y^2$ subject to the constraint $g\left(x\,y\right)\equiv x^2\+y^2-1\=0$.

Example 4.9.2

Find the extreme values of $f\left(x\,y\right)\=x y^2$ subject to the constraint $3 x\+4 y\=12$.

Example 4.9.3

Find the (shortest) distance from the origin to the plane $3x\+2 y-4 z\=7$.

Example 4.9.4

A pentagon is formed from a rectangle surmounted by an isosceles triangle.

What dimensions give the pentagon least perimeter if the area is fixed at the value 50?  

Example 4.9.5

Find the minimum distance from the point $\left(1\,-3\,2\right)$ to the plane $5x-7 y\+2 z\=3$.

Example 4.9.6

Find the point on the curve $x^2\left(y-4\right)\=1$ that is closest to the point $\left(4\,5\right)$.

Example 4.9.7

Find the point on the surface $xy\+3-z\=0$ that is closest to the point $\left(2\,7\,5\right)$.

Example 4.9.8

Example 4.9.9

Find the distance from the point $\left(1\,2\,3\right)$ to the line of intersection of the planes $5 x\+4 y-3 z\=11$ and $7 x-2 y\+6 z\=9$.

Example 4.9.10

Find the extrema of $f\left(x\,y\,z\right)\=2 x\+3 y-5 z$ subject to the constraints $3 x^2\+2 y^2\=4$ and $4 x\+3 z\=1$.