Q. 12 Given a function of n variables,... [FREE SOLUTION]

Chapter 12: Q. 12 (page 985)

Given a function of n variables, $z = f (x_{1}, x_{2}, 鈥�, x_{n}),$ and a constraint equation, $g (x_{1}, x_{2}, 鈥�, x_{n}) = 0,$ how many equations would we obtain if we tried to optimize f by the method of Lagrange multipliers?

Short Answer

Expert verified

Total $n + 1$ functions need to solve to optimize the function f,

role="math" localid="1649872718577" $\frac{鈭� f}{鈭� x_{1}} = 位 \frac{鈭� g}{鈭� x_{1}} \frac{鈭� f}{鈭� x_{2}} = 位 \frac{鈭� g}{鈭� x_{2}} 鈰� \frac{鈭� f}{鈭� x_{n}} = 位 \frac{鈭� g}{鈭� x_{n}} and g (x_{1}, x_{2}, 鈥�, x_{n}) = 0$

Step by step solution

Step 1. Given information.

Given n variables function is $z = f (x_{1}, x_{2}, 鈥�, x_{n}),$

Given constraint equation is $g (x_{1}, x_{2}, 鈥�, x_{n}) = 0 .$

Step 2. Optimization of function.

The function $z = f (x_{1}, x_{2}, 鈥�, x_{n})$ can be optimized by solving the equation $鈭� f (x_{1}, x_{2}, 鈰�, x_{n}) = 位鈭� g (x_{1}, x_{2}, 鈰�, x_{n})$ for $位, x_{1}, x_{2}, 鈰�, x_{n}$

equations that need to solve are the following.

$\frac{鈭� f}{鈭� x_{1}} = 位 \frac{鈭� g}{鈭� x_{1}} \frac{鈭� f}{鈭� x_{2}} = 位 \frac{鈭� g}{鈭� x_{2}} 鈰� \frac{鈭� f}{鈭� x_{n}} = 位 \frac{鈭� g}{鈭� x_{n}}$

and constraint equation $g (x_{1}, x_{2}, 鈥�, x_{n}) = 0 .$

So $n + 1$ functions need to solve to optimize the function f.

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91影视

Given a function of n variables, $z = f (x_{1}, x_{2}, 鈥�, x_{n}),$ and a constraint equation, $g (x_{1}, x_{2}, 鈥�, x_{n}) = 0,$ how many equations would we obtain if we tried to optimize f by the method of Lagrange multipliers?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Optimization of function.

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91影视

Given a function of n variables, z=fx1,x2,鈥�,xn,and a constraint equation, gx1,x2,鈥�,xn=0,how many equations would we obtain if we tried to optimize f by the method of Lagrange multipliers?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Optimization of function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Decision Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

Given a function of n variables, $z = f (x_{1}, x_{2}, 鈥�, x_{n}),$ and a constraint equation, $g (x_{1}, x_{2}, 鈥�, x_{n}) = 0,$ how many equations would we obtain if we tried to optimize f by the method of Lagrange multipliers?