# Tagged Questions

For questions on Lagrange multipliers, a strategy to solve constrained optimisation problems.

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### A maximization problem

I'm trying to find the maximum value of the function $f(x,y)=(ax+by)^p+x^p$ subject to the constraint $x^p+y^p=1$. Here, $a,b$ and $p$ are constants with $a,b>0$ and $p>1$, and $x,y>0$. I ...
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### Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?

Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$? I've been attempting this with Lagrange multipliers in a few different ways. However, the ...
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### Find the Lagrange multipliers with one constraint: $f(x,y,z) = xyz$ and $g(x,y,z) = x^2+2y^2+3z^2 = 6$

Where $f(x,y,z) = xyz$ and the constraint is $g(x,y,z) = x^2+2y^2+3z^2 = 6$ I have tried this problem like three or four times and not gotten the solution, I even asked this question once and got the ...
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### Minimize Frobenius norm with unitary constraint

I am trying to find a unitary tramsformation, $M$, that minimizes $\Vert MA-B \Vert_F^2$ where $A$ and $B$ are $N\times L,\;L\ge N$. I know how to solve it without the unitary constraint. I thought ...
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### Clarification on optimization problem, continued

Background This is a follow-up to this question. The problem statement is the same: Maximize $$f(\alpha_1, \dots, \alpha_5) = \sum_{1 \le i < j < k \le 5} \alpha_i \alpha_j \alpha_k$$ ...
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### Using the method of Lagrange multipliers, find the extreme values of a function

Using the method of Lagrange multipliers, find the extreme values of the function $f(x,y)= \frac{2y^3}{3} + 2x^2 +1$ on the ellipse $5x^2 + y^2 = 1/9$ . Identify the (absolute) maximal and minimal ...
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### In Lagrange Multiplier, why level curves of $f$ and $g$ are tangent to each other?

In Lagrange multiplier method, e.g. optimize a function $f(x_1, \dots, x_n)$ under a constraint $g(x_1, \dots, x_n) = 0$. There is a fact that $\nabla f$ is parallel to $\nabla g$ which is given rise ...
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### Finding the distance between a point and a parabola with different methods

I'm trying to find the shortest distance from point $(3,0)$ to the parabola $y= x^2$ using the method of Lagrange Multipliers (my practice), and by "reducing to unstrained problem in one variable" (...
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### Determine the points where $f$ is has a local minimum/maximum. Multivariable calculus question.

This is not homework, but it is in my book and I find it hard to solve: Determine the points where $f$ is has a local minimum/maximum. Determine if it strong/weak and absolute/relative and ...
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### First derivative of Lagrange polynomial

Given the Lagrange basis polynomial as: $L_i(x)= \prod_{m=0, m \neq i}^n \frac{x-x_m}{x_i-x_m}$ is there a generic equation for the first derivative ${L_i}'(x)$ for any order,t hat is for any $n$?
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### Max and Min using Lagrange Multipliers

Suppose A is a symmetric matrix. Show that the maximum and minimum of $\mathbf x ^T A \mathbf x$ subject to the constraint $\mathbf x ^T \mathbf x=1$ are the maximum and minimum eigenvalues of A. I ...
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### Use Lagrange Multipliers to find the absolute extrema

Use Lagrange Multipliers to find the absolute extrema (if any) of: $f(x,y) = 4x^2 + 9y^2$; subject to $2x +3y = 6$. Using Lagrange I end up with one point: $(\frac{3}{2}, 1)$ I'm just not sure how ...
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### Related Methods: Lagrange Multipliers

It really pains me to ask this question, but I was working on an optimization problem and wanted to show a friend how we could also use Lagrange Multipliers to solve it. I was considering the ...
### Method to find the extremal values of $xyz$ subject to $x^2+2y^2+3z^2=a$
This question has been asked before but I want to lay out my method and get feedback on reasoning and process this took me a long to put together as I am new to the formatting: Let the function $f$ ...