Lagrange multipliers for functions of two variables. Hello friends, today it’s about Lagrange multipliers for functions of two independent variables. Have a look!!

**Lagrange multipliers for functions of two variables**

**Solved example on Lagrange multipliers for functions ****of two variables**

Disclaimer: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the example.

**Example **

According to Stroud and Booth (2011)* “Use Lagrange’s method of undetermined multipliers to obtain the stationary value of the following function subject to the constraint

”

**Solution**

Now here the given function is

(1)

(2)

First of all, I’ll get the values of and .

**Step 1**

As I can see, both and are functions of two variables and . So I’ll use the method for the * first-order partial derivatives of functions with two variables*.

Thus I’ll differentiate equation (1) partially with respect to to get

Next, I’ll differentiate equation (1) partially with respect to to get

Similarly, I’ll differentiate equation (2) partially with respect to to get

Then I’ll differentiate equation (2) partially with respect to to get

Now I’ll form and solve two different equations

together with the equation (2).

So from I get . And this means

(3)

Also from , I get So this means

(4)

Now I’ll solve the equations (3) and (4) to get the values of and .

**Step 2**

So I’ll add equations (3) and (4) to get

Next, I’ll simplify it to get

Thus either or . So I’ll choose since I’ve to get the values of and .

If , then . Next, I’ll substitute in equation (2). And that gives

So this means

Therefore I can say that

Since , I can say that the value of is

Next, I’ll substitute these values of and in equation (1) to get the value of as

Now I’ll simplify it to get

Therefore I can say that the stationary value of the following function subject to the constraint is Also, the value of the function is .

Hence I can conclude that this is the solution to the given example.

Dear friends, this is the end of today’s post on Lagrange multipliers for functions of two independent variables. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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