ยง14.8 Lagrange Multipliers

Lecture notes

Lagrange multipliers

Examples

  1. Find the point(s) on the curve \[ x^2y = 2 \] closest to the origin.
  2. Find the point(s) on the surface \[ xyz = 1 \] closest to the origin.
  3. Suppose the Celsius temperature of a point on the sphere \[ x^2 + y^2 + z^2 = 1 \] is given by \[ T = 20xyz^2. \] Find the point(s) of highest temperature.

Multiple constraints

Examples

  1. The plane \[ x - y + z = 0 \] cuts the cylinder \[ y^2 + z^2 = 1 \] in an ellipse. Find the points on the ellipse closest to and farthest from the origin.