# Need help with Lagrange multipliers

• Tiome_nguyen
In summary, the conversation was about two problems in multivariable calculus homework. The first problem involved finding the directional derivatives of a function at a given point and finding the equation of the tangent plane to the function at that point. The second problem involved using Lagrange multipliers to find the maximum and minimum values of a function subject to certain constraints, and then finding the absolute maximum and minimum values in a bounded region. The poster requested help with these problems and mentioned that they had spent an hour trying to solve one of them.

#### Tiome_nguyen

Hello everyone, i have 2 problems in my multivariable calculus homework that i can't solve . Please help me out, thank you so much

1/f(x,y)= [(x^2) -2y]^(0.5)
a) Find directional derivatives of f at (2,-6) in the direction of <-4,3>
b) Find equation of the tangent plane to the function f(x,y) in problem 1 at the point (2,-6)
i got the part a) and but i have no idea how to do part b)

2/
a)Use Lagrange multipliers to find max and min of function P(x,y) = x(y^2) +(x^3)y-5xy subject to the constraint boundaries xy = 4 , y = x+3 , y = x-3
b) find absolute maximum and minimum on the region bounded by xy <= 4, x-3 <= y <= x+3 .
i spent an hour for this problem but couldn't solve for it

Tiome_nguyen said:
Hello everyone, i have 2 problems in my multivariable calculus homework that i can't solve . Please help me out, thank you so much

1/f(x,y)= [(x^2) -2y]^(0.5)
a) Find directional derivatives of f at (2,-6) in the direction of <-4,3>
b) Find equation of the tangent plane to the function f(x,y) in problem 1 at the point (2,-6)
i got the part a) and but i have no idea how to do part b)

2/
a)Use Lagrange multipliers to find max and min of function P(x,y) = x(y^2) +(x^3)y-5xy subject to the constraint boundaries xy = 4 , y = x+3 , y = x-3
b) find absolute maximum and minimum on the region bounded by xy <= 4, x-3 <= y <= x+3 .
i spent an hour for this problem but couldn't solve for it

Hello Timoe_nguyen and welcome to the forums.

When we give out help here on PF we ask the posters to show any work that they have done before giving them advice. The reason we do this is because if people were just given answers then no-one benefits since the student will not understand for themselves, and also because we need to see the thought process of the individual so that we can correct any misconceptions they may have.

For question 1, what is the definition of the directional derivative and given a surface in 3-space, how do you define a plane with two linearly independent vectors that lie on the plane?

For question 2, again what is the definition of a Lagrange Multiplier, and in terms of minimums and maximums, what does this mean when we use inequalities?

## 1. What are Lagrange multipliers?

Lagrange multipliers are a mathematical technique used to find the maximum or minimum value of a function subject to a set of constraints. They are commonly used in optimization problems and involve finding the stationary points of a function.

## 2. When should Lagrange multipliers be used?

Lagrange multipliers should be used when solving optimization problems with constraints. They are especially useful when the constraints are non-linear or when there are multiple constraints that need to be considered simultaneously.

## 3. How do Lagrange multipliers work?

Lagrange multipliers work by introducing a new variable, known as the Lagrange multiplier, to the objective function. This allows the constraints to be incorporated into the objective function, creating a new function with the same stationary points as the original function but with the constraints satisfied.

## 4. What is the formula for Lagrange multipliers?

The formula for Lagrange multipliers is ∇f(x,y) = λ∇g(x,y), where f(x,y) is the objective function, g(x,y) is the constraint function, and λ is the Lagrange multiplier. This formula is used to find the stationary points of the function and the corresponding values of λ.

## 5. Are there any limitations or drawbacks to using Lagrange multipliers?

One limitation of using Lagrange multipliers is that they can only be used for optimization problems with constraints. They also require the constraints to be differentiable, which may not always be the case. Additionally, the Lagrange multiplier may not have a physical interpretation, making it difficult to interpret the results in some cases.