Find the maximum and minimum values of the function f(x,y,z)=yz+xy
 subject to the constraints y2+z2=225
 and  xy=7

We need to find the maximum and minimum values of the function:

\[
f(x, y, z) = yz + xy
\]

subject to the constraints:

\[
y^2 + z^2 = 225 \quad \text{(1)}
\]
\[
xy = 7 \quad \text{(2)}
\]

To solve this problem, we will apply the method of **Lagrange multipliers**, which involves introducing multipliers \(\lambda\) and \(\mu\) for the two constraints.

### Step 1: Define the Lagrange function
The Lagrange function \( \mathcal{L}(x, y, z, \lambda, \mu) \) is given by:

\[
\mathcal{L}(x, y, z, \lambda, \mu) = f(x, y, z) - \lambda(g_1(y, z) - 225) - \mu(g_2(x, y) - 7)
\]

where:
- \( f(x, y, z) = yz + xy \)
- \( g_1(y, z) = y^2 + z^2 \)
- \( g_2(x, y) = xy \)

So the Lagrange function becomes:

\[
\mathcal{L}(x, y, z, \lambda, \mu) = yz + xy - \lambda(y^2 + z^2 - 225) - \mu(xy - 7)
\]

### Step 2: Take partial derivatives and set them equal to zero

We need to take the partial derivatives of \( \mathcal{L} \) with respect to \( x \), \( y \), \( z \), \( \lambda \), and \( \mu \), and set each to zero.

#### Partial derivatives:
\[
\frac{\partial \mathcal{L}}{\partial x} = y - \mu y = 0 \quad \Rightarrow \quad y(1 - \mu) = 0
\]
\[
\frac{\partial \mathcal{L}}{\partial y} = z + x - 2\lambda y - \mu x = 0
\]
\[
\frac{\partial \mathcal{L}}{\partial z} = y - 2\lambda z = 0
\]
\[
\frac{\partial \mathcal{L}}{\partial \lambda} = -(y^2 + z^2 - 225) = 0 \quad \Rightarrow \quad y^2 + z^2 = 225
\]
\[
\frac{\partial \mathcal{L}}{\partial \mu} = -(xy - 7) = 0 \quad \Rightarrow \quad xy = 7
\]

### Step 3: Solve the system of equations

From \( \frac{\partial \mathcal{L}}{\partial x} = 0 \), we have \( y(1 - \mu) = 0 \). This gives two cases:

#### Case 1: \( y = 0 \)

- If \( y = 0 \), then from the constraint \( xy = 7 \), we get \( x = \frac{7}{y} \), but this is undefined, so \( y = 0 \) leads to an invalid solution.

#### Case 2: \( \mu = 1 \)

- From the equation \( y(1 - \mu) = 0 \), we get \( \mu = 1 \).


Find the maximum and minimum values of the function f(x, y, z) = yz + xy subject to the constraints y^2 + z^2 = 225 and xy = 7.

Solve the problem of finding the maximum and minimum values of the function f(x, y, z) = yz + xy, subject to the constraints y² + z² = 225 and xy = 7, using Lagrange multipliers. This problem covers advanced multivariable calculus concepts and is suitable for college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation