Section 14.8: Lagrange Multipliers

In a previous example, we maximized a volume function \(V = xyz\) subject to the constraint \(2xz + 2yz + xy = 12\), which expressed the side condition that the surface area was 12 m\(^2\).

In this section we present Lagrange’s method for maximizing or minimizing a general function \(f(x, y, z)\) subject to a constraint (or side condition) of the form \(g(x, y, z) = k\).

It’s easier to explain the geometric basis of Lagrange’s method for functions of two variables. So we start by trying to find the extreme values of \(f(x, y)\) subject to a constraint of the form \(g(x, y) = k\). In other words, we seek the extreme values of \(f(x, y)\) when the point \((x, y)\) is restricted to lie on the level curve \(g(x, y) = k\).

Figure 1 shows this curve together with several level curves of \(f\). These have the equations \(f(x, y) = c\), where \(c = 7, 8, 9, 10, 11\). To maximize \(f(x, y)\) subject to \(g(x, y) = k\) is to find the largest value of \(c\) such that the level curve \(f(x, y) = c\) intersects \(g(x, y) = k\). It appears from Figure 1 that this happens when these curves just touch each other, that is, when they have a common tangent line. (Otherwise, the value of \(c\) could be increased further.) This means that the normal lines at the point \((x_0, y_0)\) where they touch are identical. So the gradient vectors are parallel; that is, \(\nabla f(x_0, y_0) = \lambda \nabla g(x_0, y_0)\) for some scalar \(\lambda\).

This kind of argument also applies to the problem of finding the extreme values of \(f(x, y, z)\) subject to the constraint \(g(x, y, z) = k\). Thus the point \((x, y, z)\) is restricted to lie on the level surface \(S\) with equation \(g(x, y, z) = k\). Instead of the level curves in Figure 1, we consider the level surfaces \(f(x, y, z) = c\) and argue that if the maximum value of \(f\) is \(f(x_0, y_0, z_0) = c\), then the level surface \(f(x, y, z) = c\) is tangent to the level surface \(g(x, y, z) = k\) and so the corresponding gradient vectors are parallel.

This intuitive argument can be made precise as follows. Suppose that a function \(f\) has an extreme value at a point \(P(x_0, y_0, z_0)\) on the surface \(S\) and let \(C\) be a curve with vector equation \(\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle\) that lies on \(S\) and passes through \(P\). If \(t_0\) is the parameter value corresponding to the point \(P\), then \(\mathbf{r}(t_0) = \langle x_0, y_0, z_0 \rangle\). The composite function \(h(t) = f(x(t), y(t), z(t))\) represents the values that \(f\) takes on the curve \(C\). Since \(f\) has an extreme value at \((x_0, y_0, z_0)\), it follows that \(h\) has an extreme value at \(t_0\), so \(h'(t_0) = 0\). But if \(f\) is differentiable, we can use the Chain Rule to write \[ 0 = h'(t_0) = f_x(x_0, y_0, z_0)x'(t_0) + f_y(x_0, y_0, z_0)y'(t_0) + f_z(x_0, y_0, z_0)z'(t_0) = \nabla f(x_0, y_0, z_0) \cdot \mathbf{r}'(t_0) \] This shows that the gradient vector \(\nabla f(x_0, y_0, z_0)\) is orthogonal to the tangent vector \(\mathbf{r}'(t_0)\) to every such curve \(C\). But we already know that the gradient vector of \(g\), \(\nabla g(x_0, y_0, z_0)\), is also orthogonal to \(\mathbf{r}'(t_0)\) for every such curve. This means that the gradient vectors \(\nabla f(x_0, y_0, z_0)\) and \(\nabla g(x_0, y_0, z_0)\) must be parallel. Therefore, if \(\nabla g(x_0, y_0, z_0) \neq \mathbf{0}\), there is a number \(\lambda\) such that \[ \nabla f(x_0, y_0, z_0) = \lambda \nabla g(x_0, y_0, z_0) \] The number \(\lambda\) in Equation 1 is called a Lagrange multiplier. The procedure is described the next page.

Method of Lagrange Multipliers To find the maximum and minimum values of \(f(x, y, z)\) subject to the constraint \(g(x, y, z) = k\) [assuming that these extreme values exist and \(\nabla g \neq \mathbf{0}\) on the surface \(g(x, y, z) = k\)]: (a) Find all values of \(x, y, z\), and \(\lambda\) such that \[ \nabla f(x, y, z) = \lambda \nabla g(x, y, z) \] and \[ g(x, y, z) = k \] (b) Evaluate \(f\) at all the points \((x, y, z)\) that result from step (a). The largest of these values is the maximum value of \(f\); the smallest is the minimum value of \(f\).

If we write the vector equation \(\nabla f = \lambda \nabla g\) in terms of components, then the equations in step (a) become \[ f_x = \lambda g_x \quad f_y = \lambda g_y \quad f_z = \lambda g_z \quad g(x, y, z) = k \] This is a system of four equations in the four unknowns \(x, y, z\), and \(\lambda\), but it is not necessary to find explicit values for \(\lambda\).

For functions of two variables the method of Lagrange multipliers is similar to the method just described. To find the extreme values of \(f(x, y)\) subject to the constraint \(g(x, y) = k\), we look for values of \(x, y\), and \(\lambda\) such that \[ \nabla f(x, y) = \lambda \nabla g(x, y) \quad \text{and} \quad g(x, y) = k \] This amounts to solving three equations in three unknowns: \[ f_x = \lambda g_x \quad f_y = \lambda g_y \quad g(x, y) = k \]

EXAMPLE 1 A rectangular box without a lid is to be made from 12 m\(^2\) of cardboard. Find the maximum volume of such a box.

EXAMPLE 2 Find the extreme values of the function \(f(x, y) = x^2 + 2y^2\) on the circle \(x^2 + y^2 = 1\).

EXAMPLE 3 Find the extreme values of \(f(x, y) = x^2 + 2y^2\) on the disk \(x^2 + y^2 \le 1\).

EXAMPLE 4 Find the points on the sphere \(x^2 + y^2 + z^2 = 4\) that are closest to and farthest from the point \((3, 1, -1)\).

Suppose now that we want to find the maximum and minimum values of a function \(f(x, y, z)\) subject to two constraints (side conditions) of the form \(g(x, y, z) = k\) and \(h(x, y, z) = c\). Geometrically, this means that we are looking for the extreme values of \(f\) when \((x, y, z)\) is restricted to lie on the curve of intersection \(C\) of the level surfaces \(g(x, y, z) = k\) and \(h(x, y, z) = c\). Suppose \(f\) has such an extreme value at a point \(P(x_0, y_0, z_0)\). We know that \(\nabla f\) is orthogonal to \(C\) at \(P\). But we also know that \(\nabla g\) and \(\nabla h\) are orthogonal to \(C\) at \(P\). This means that the gradient vector \(\nabla f(x_0, y_0, z_0)\) is in the plane determined by \(\nabla g(x_0, y_0, z_0)\) and \(\nabla h(x_0, y_0, z_0)\). (We assume that these gradient vectors are not zero and not parallel.) So there are numbers \(\lambda\) and \(\mu\) (called Lagrange multipliers) such that \[ \nabla f(x_0, y_0, z_0) = \lambda \nabla g(x_0, y_0, z_0) + \mu \nabla h(x_0, y_0, z_0) \] In this case Lagrange’s method is to look for extreme values by solving five equations in the five unknowns \(x, y, z, \lambda,\) and \(\mu\). These equations are obtained by writing Equation 16 in terms of its components and using the constraint equations: \[ f_x = \lambda g_x + \mu h_x \] \[ f_y = \lambda g_y + \mu h_y \] \[ f_z = \lambda g_z + \mu h_z \] \[ g(x, y, z) = k \] \[ h(x, y, z) = c \]

EXAMPLE 5 Find the maximum value of the function \(f(x, y, z) = x + 2y + 3z\) on the curve of intersection of the plane \(x - y + z = 1\) and the cylinder \(x^2 + y^2 = 1\).

Pictured are a contour map of \(f\) and a curve with equation \(g(x, y) = 8\). Estimate the maximum and minimum values of \(f\) subject to the constraint that \(g(x, y) = 8\). Explain your reasoning.

1. Use a graphing calculator or computer to graph the curve with equation \(x^3 + y^3 = 3xy\). At what point on the curve does it seem that the tangent line is horizontal?
2. Use the method of Lagrange multipliers to find the exact coordinates of the point in part (a).

Find the extreme values of \(f(x, y) = x^2 + y^2\) subject to the constraint \(xy = 1\).

Find the extreme values of \(f(x, y) = xy\) subject to the constraint \(x^2 + y^2 = 1\).

Find the extreme values of \(f(x, y) = x^2 + 2y^2\) subject to the constraint \(x^2 + y^2 = 1\).

Find the extreme values of \(f(x, y) = x^2 - y^2\) subject to the constraint \(x^2 + y^2 = 1\).

Find the extreme values of \(f(x, y, z) = 2x + 2y + z\) subject to the constraint \(x^2 + y^2 + z^2 = 9\).

Find the extreme values of \(f(x, y, z) = x^2 + y^2 + z^2\) subject to the constraint \(x + y + z = 12\).

Find the extreme values of \(f(x, y, z) = xyz\) subject to the constraint \(x^2 + 2y^2 + 3z^2 = 6\).

Find the extreme values of \(f(x, y, z) = x^2y^2z^2\) subject to the constraint \(x^2 + y^2 + z^2 = 1\).

Find the extreme values of \(f(x, y, z) = x^4 + y^4 + z^4\) subject to the constraint \(x^2 + y^2 + z^2 = 1\).

Find the extreme values of \(f(x, y, z, t) = x + y + z + t\) subject to the constraint \(x^2 + y^2 + z^2 + t^2 = 1\).

Find the extreme values of \(f(x_1, x_2, ..., x_n) = x_1 + x_2 + ... + x_n\) subject to the constraint \(x_1^2 + x_2^2 + ... + x_n^2 = 1\).

Find the extreme values of \(f(x_1, x_2, ..., x_n) = x_1^2 + x_2^2 + ... + x_n^2\) subject to the constraint \(x_1 + x_2 + ... + x_n = 1\).

The method of Lagrange multipliers assumes that the extreme values exist, but that is not always the case. Show that the problem of finding the minimum value of \(f(x, y) = x\) subject to the constraint \(y^2 + x^4 - x^3 = 0\) can be solved using Lagrange multipliers, but \(f\) does not have a maximum value with that constraint.

Find the maximum value of \(f(x, y) = e^{xy}\) subject to the constraint \(x^3 + y^3 = 16\). Show that \(f\) has no minimum value with this constraint.

Find the extreme values of \(f(x, y) = x^2 + y^2\) subject to both constraints \(x^2 + y^2 = 1\) and \(x + y = 1\).

Find the extreme values of \(f(x, y, z) = x^2 + y^2 + z^2\) subject to both constraints \(x^2 + y^2 + z^2 = 1\) and \(x + y + z = 1\).

Find the extreme values of \(f(x, y, z) = yz + xy\) subject to both constraints \(xy = 1\) and \(y^2 + z^2 = 1\).

Find the extreme values of \(f(x, y, z) = x^2 + y^2 + z^2\) subject to both constraints \(z^2 = x^2 + y^2\) and \(x + y - z = -1\).

Find the absolute maximum and minimum values of \(f(x, y) = x^2y\) on the ellipse \(x^2 + 2y^2 = 6\).

Find the absolute maximum and minimum values of \(f(x, y) = x^2 + 4x + y^2 - 4y\) on the disk \(x^2 + y^2 \le 9\).

Find the absolute maximum and minimum values of \(f(x, y) = 2x^2 + 3y^2 - 4x - 5\) on the disk \(x^2 + y^2 \le 16\).

Find the absolute maximum and minimum values of \(f(x, y) = e^{-x^2-y^2}(x^2 + 2y^2)\) on the disk \(x^2 + y^2 \le 4\).

Consider the problem of maximizing the function \(f(x, y) = 2x + 3y\) subject to the constraint \(\sqrt{x} + \sqrt{y} = 5\). (a) Try using Lagrange multipliers to solve the problem. (b) Does \(f(25, 0)\) give a larger value than the one in part (a)? (c) Solve the problem by graphing the constraint curve and several level curves of \(f\). (d) Explain why the method of Lagrange multipliers fails to solve the problem. (e) What is the significance of \(f(9, 4)\)?

1. Use a computer algebra system to solve the system of equations that arises in using Lagrange multipliers to maximize \(f(x, y, z) = x^2y^2z^2\) subject to the constraint \(x^2 + y^2 + z^2 = 1\).
2. Solve the problem in part (a) with the aid of a graph and level surfaces. Use your CAS to solve the equations numerically. Compare your answers with those in part (a).

The total production P of a certain product depends on the amount L of labor used and the amount K of capital investment. In Sections 14.1 and 14.3 we discussed how the Cobb-Douglas model \(P = bL^\alpha K^{1-\alpha}\) follows from certain economic assumptions, where b and \(\alpha\) are positive constants and \(\alpha < 1\). If the cost of a unit of labor is m and the cost of a unit of capital is n, and the company can spend only p dollars as its total budget, then maximizing the production P is subject to the constraint \(mL + nK = p\). Show that the maximum production occurs when \(L = \frac{\alpha p}{m}\) and \(K = \frac{(1-\alpha)p}{n}\)

Referring to Exercise 27, we now suppose that the production is fixed at \(bL^\alpha K^{1-\alpha} = Q\), where Q is a constant. What values of L and K minimize the cost function \(C(L, K) = mL + nK\)?

Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral. [Hint: Use Heron’s formula for the area: \(A = \sqrt{s(s-x)(s-y)(s-z)}\), where \(s = p/2\) and \(x, y, z\) are the lengths of the sides.]

Find the shortest distance from the point \((2, 1, -1)\) to the plane \(x + y - z = 1\).

Find the points on the cone \(z^2 = x^2 + y^2\) that are closest to the point \((4, 2, 0)\).

Find the point on the sphere \(x^2 + y^2 + z^2 = 4\) that is closest to the point \((3, 1, -1)\).

Find the point on the sphere \(x^2 + y^2 + z^2 = 1\) that is closest to the point \((2, 1, 2)\).

Find the points on the surface \(y^2 = 9 + xz\) that are closest to the origin.

Find the points on the surface \(x^2y^2z = 1\) that are closest to the origin.

Find the dimensions of the box with volume 1000 cm\(^3\) that has minimal surface area.

Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm\(^2\).

Find the maximum and minimum volumes of a rectangular box whose surface area is 1500 cm\(^2\) and whose total edge length is 200 cm.

The plane \(x + y + 2z = 2\) intersects the paraboloid \(z = x^2 + y^2\). Find the points on this ellipse that are nearest to and farthest from the origin.

The plane \(4x - 3y + 8z = 5\) intersects the cone \(z^2 = x^2 + y^2\). Find the points on this ellipse that are nearest to and farthest from the origin.

Find the maximum and minimum values of \(f(x, y, z) = x^2 + y^2 + z^2\) subject to the given constraints. Use a computer algebra system to solve the system of equations that arises in using Lagrange multipliers. (If your CAS finds only one solution, you may need to use additional commands.) \(f(x, y, z) = 3x + 2y + 4z\); \(x^2 + 2y^2 + 6z^2 = 1\)

Find the maximum and minimum values of \(f(x, y, z) = x^2 + y^2 + z^2\) subject to the given constraints. Use a computer algebra system to solve the system of equations that arises in using Lagrange multipliers. (If your CAS finds only one solution, you may need to use additional commands.) \(f(x, y, z) = x^2 + y^2 + z^2\); \(x^4 + y^4 + z^4 = 1\)

Find the maximum value of \(f(x_1, x_2, ..., x_n) = \sqrt[n]{x_1 x_2 ... x_n}\) given that \(x_1, x_2, ..., x_n\) are positive numbers and \(\sum_{i=1}^n x_i = c\), where c is a constant. Deduce from part (a) that if \(x_1, x_2, ..., x_n\) are positive numbers, then \(\sqrt[n]{x_1 x_2 ... x_n} \le \frac{x_1 + x_2 + ... + x_n}{n}\) This inequality says that the geometric mean of n numbers is no larger than the arithmetic mean of the numbers. Under what circumstances are these two means equal?

1. Maximize \(\sum_{i=1}^n x_i y_i\) subject to the constraints \(\sum_{i=1}^n x_i^2 = 1\) and \(\sum_{i=1}^n y_i^2 = 1\).
2. Put \(x_i = \frac{a_i}{\sqrt{\sum a_j^2}}\) and \(y_i = \frac{b_i}{\sqrt{\sum b_j^2}}\) to show that \(\sum a_i b_i \le \sqrt{\sum a_i^2} \sqrt{\sum b_i^2}\) for any numbers \(a_1, ..., a_n, b_1, ..., b_n\). This inequality is known as the Cauchy-Schwarz Inequality.