Study guide and
5 practice problems
on:
Lagrange multipliers
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Lagrange multipliers are a convenient tool to solve constrained minimization problems.
To use Lagrange multipliers to solve the problem $$\min f(x,y,z) \text{ subject to } g(x,y,z) = 0,$$
Form the augmented function $$L(x,y,z,\lambda) = f(x,y,z) - \lambda g(x,y,z)$$
Set all partial derivatives of $L$ equal to zero
Solve for $x,y,z$.
Lagrange multipliers also work when solving a constrained maximization problem.
Related topics
Max/min problems
(8 problems)
Multivariable calculus
(147 problems)
Practice problems
Of all the rectangles of perimeter P, which has the largest area?
Solution
Of the rectangular prisms with surface area A, which has maximal volume?
Solution
Consider an open box with no top, as shown. The box has volume $32$ and dimensions $x,y,z$. Using Lagrange multipliers, find the dimensions of the box with minimal surface area.
Solution
Use Lagrange multipliers to find the triangle of largest area that can be inscribed in a circle of radius $r$.
Solution
Suppose $x+ y+z=A$. What values of $x,y,z$ are such that $x^2 + y^2 + z^2$ is the smallest?
Solution