14.8 Videos Guide

14.8a

• The method of Lagrange multipliers

  • In $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$, when optimizing a function $f$ subject to the equation $g=c$, we use the fact that $\mathbf{\nabla}f=\lambda\mathbf{\nabla}g$ ($\lambda$ is called the Lagrange multiplier)

Exercises:

• Find the maximum area for a rectangle inscribed in the ellipse $\frac{x^{2}}{3^{2}}+\frac{y^{2}}{4^{2}}$

14.8b

• Use Lagrange multipliers to find the extreme values of the function subject to the given constraint.

  • $f(x,\ y)=3x+y$; $x^{2}+y^{2}=10$

  • $f(x,\ y)=xe^{y}$; $x^{2}+y^{2}=2$

14.8c

• $f(x,\ y,\ z)=\ln\left(x^{2}+1\right)+\ln\left(y^{2}+1\right)+ln(z^{2}+1)$; $x^{2}+y^{2}+z^{2}=12$

14.8d

• Use Lagrange multipliers to find the points on the cone $z^{2}=x^{2}+y^{2}$ that are closest to the point $(4,\ 2,\ 0)$.

14.8e

• When optimizing a function $f(x,\ y,\ z)$ subject to two constraints $g=k$ and
  $h=c$, we use $\mathbf{\nabla}f=\lambda\mathbf{\nabla}g+\mu\mathbf{\nabla}h$

Exercise:

• Find the extreme values of $f$ subject to both constraints.
  $f(x,\ y,\ z)=z$; $x^{2}+y^{2}=z^{2}$, $x+y+z=24$