Problem 2
Question
If \(f(x, y)=x^{2}+y^{2}\) and \(g(x, y)=2 x+3 y-4=0,\) write the Lagrange multiplier conditions that must be satisfied by a point that maximizes or minimizes \(f\) subject to \(g(x, y)=0\).
Step-by-Step Solution
Verified Answer
Answer: The Lagrange multiplier conditions are \(2x - 2\lambda = 0\), \(2y - 3\lambda = 0\), and \(-2x - 3y + 4 = 0\).
1Step 1: Introduce the Lagrange function
The Lagrange function \(L(x, y, \lambda)\) is given by
$$ L(x, y, \lambda) = f(x, y) - \lambda g(x, y) = x^2 + y^2 - \lambda(2x + 3y - 4). $$
2Step 2: Find the partial derivatives of the Lagrange function
Calculate the partial derivatives of the Lagrange function with respect to \(x, y\), and \(\lambda\).
$$
\frac{\partial L}{\partial x} = 2x - 2\lambda, \;\;
\frac{\partial L}{\partial y} = 2y - 3\lambda, \;\;
\frac{\partial L}{\partial \lambda} = -2x - 3y + 4.
$$
3Step 3: Set these derivatives equal to zero
To maximize or minimize the function \(f(x, y)\) subject to the constraint \(g(x, y) = 0\), set the partial derivatives of the Lagrange function equal to zero.
$$
2x - 2\lambda = 0,\;\;
2y - 3\lambda = 0, \;\;
-2x - 3y + 4 = 0.
$$
These are the Lagrange multiplier conditions that must be satisfied by a point that maximizes or minimizes \(f(x, y)\) subject to \(g(x, y) = 0\).
Other exercises in this chapter
Problem 1
A function is defined by \(z=x^{2} y-x y^{2} .\) Identify the independent and dependent variables.
View solution Problem 1
Suppose you are standing on the surface \(z=f(x, y)\) at the point \((a, b, f(a, b)) .\) Interpret the meaning of \(f_{x}(a, b)\) and \(f_{y}(a, b)\) in terms o
View solution Problem 2
Describe the usual appearance of a smooth surface at a saddle point.
View solution Problem 2
Explain why \(f(x, y)\) must approach \(L\) as \((x, y)\) approaches \((a, b)\) along all paths in the domain in order for $$\lim _{(x, y) \rightarrow(a, b)} f(
View solution