Questions tagged [lagrange-multiplier]

When are Lagrange multipliers useful?

One of the most common problems in calculus is that of finding maxima or minima (in general, "extrema") of a function, but it is often difficult to find a closed form for the function being extremized. Such difficulties often arise when one wishes to maximize or minimize a function subject to fixed outside conditions or constraints. The method of Lagrange multipliers is a powerful tool for solving this class of problems without the need to explicitly solve the conditions and use them to eliminate extra variables.

Put more simply, it's usually not enough to ask, "How do I minimize the aluminum needed to make this can?" (The answer to that is clearly "Make a really, really small can!") You need to ask, "How do I minimize the aluminum while making sure the can will hold $10$ ounces of soup?" Or similarly, "How do I maximize my factory's profit given that I only have $\$15,000$ to invest?" Or to take a more sophisticated example, "How quickly will the roller coaster reach the ground assuming it stays on the track?" In general, Lagrange multipliers are useful when some of the variables in the simplest description of a problem are made redundant by the constraints.

The mathematics of Lagrange multipliers:

To find critical points of a function $f(x, y, z)$ on a level surface $g(x, y, z) = C$ (or subject to the constraint $g(x, y, z) = C$), we must solve the following system of simultaneous equations: $$∇f(x, y, z) = λ∇g(x, y, z)$$ $$g(x, y, z) = C$$ Remembering that $∇f$ and $∇g$ are vectors, we can write this as a collection of four equations in the four unknowns $x, y, z,$ and $λ$ : $$f_x(x, y, z) = λ~g_x(x, y, z)$$ $$f_y(x, y, z) = λ~g_y(x, y, z)$$ $$fz(x, y, z) = λ~g_z(x, y, z)$$ $$g(x, y, z) = C$$ The variable $~λ~$ is a dummy variable called a Lagrange multiplier; we only really care about the values of $x, y,$ and $z$ .

Once we have found all the critical points, we plug them into $~f~$ to see where the maxima and minima. The critical points where $~f~$ is greatest are maxima and the critical points where $~f~$ is smallest are minima.

Note:

Solving the system of equations can be hard! Here are some tricks that may help:

$1.\quad$ Since we don’t actually care what $~λ~$ is, you can first solve for $~λ~$ in terms of $~x,~ y,~$ and $~z~$ to remove $~λ~$ from the equations.

$2.\quad$ Try first solving for one variable in terms of the others.

$3.\quad$ Remember that whenever you take a square root, you must consider both the positive and the negative square roots.

$4.\quad$ Remember that whenever you divide an equation by an expression, you must be sure that the expression is not $~0~$. It may help to split the problem into two cases: first solve the equations assuming that a variable is $~0~$, and then solve the equations assuming that it is not $~0~$.

Reference:

https://en.wikipedia.org/wiki/Lagrange_multiplier