2 votes
Accepted

Find Maximum and Minimum distance from origin to $f(x,y)$ using the Lagrange method.

You've done fine to conclude that $x^2=y^2.$ That means $x=y$ or that $x=-y.$ Plug each of those into your curve's equation. You should find four distinct points of two distinct distances from the ...
Cameron Buie's user avatar
1 vote

Find Maximum and Minimum distance from origin to $f(x,y)$ using the Lagrange method.

You need to plug $y = x$ and $y = -x$ into the equation and solve for $x$ and $y$ properly (you made a mistake when solving). Once you do that you will get $4$ possible solutions. Two will be at one ...
Zarrax's user avatar
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1 vote

Find Maximum and Minimum distance from origin to $f(x,y)$ using the Lagrange method.

Note that this method relies heavily on the AM-GM inequality,but you can connect it with your Lagrage Multiplier method to sort out the points that lie furthest from the origin. For the farthest, ...
Wang YeFei's user avatar
  • 6,538
1 vote

Maximise $f(x,y)=x^2+y^2$ on contraint that looks like infinity sign

You first define the Lagrangian $L(x,y, \lambda) = x^2+y^2 + \lambda (x^2-y^2 - (x^2+y^2)^2).$ Then, the KKT conditions are given by: $$\begin{cases} \nabla f(x^{\star}, y^{\star}) + \lambda \nabla h(...
rcescon's user avatar
  • 208
1 vote
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Proving concavity of the Lagrange dual function

You have the right intuition. Just remember that, for $t \in (0,1)$, $g$ is concave if: $$g(tx + (1-t)y) \geq tg(x) + (1-t)g(y), $$ and if $g_1, g_2$ are affine functions and $g(x)=\min\{g_1(x),g_2(x)\...
ConEd's user avatar
  • 89
1 vote

On the value of $x$ for which a point mass falls off a curve.

An alternative way. Considering the Lagrangian $$ L={m\over 2}(\dot x^2+\dot y^2)-mgy+\lambda f(x,y),\ \ \ \ f(x,y) = y+\cosh x-2=0 $$ The movement equations are $$ \cases{ m x''+\lambda \sinh x = 0\\ ...
Cesareo's user avatar
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1 vote

Multivariable Calculus - Exercise about Lagrange multipliers

OP has derived these : $$ \begin{align}y &= 2x\lambda \tag{EQ 1} \\ 2y+x &= 2y\lambda \tag{EQ 2} \\ 0 &= 2z\lambda \tag{EQ 3} \\ x^{2}+y^{2}+z^{2}-4&= 0 \tag{EQ 4} \end{align} $$ With ...
Prem's user avatar
  • 9,819
1 vote

Find the distance between tangent space and the origin

Let's assume $F(0)=0$. Let's also assume $\alpha\neq 0$, otherwise we can get arbitrarily close to the origin in $S$ by considering $F(ta)$ for arbitrarily small, positive $t$. Fix $0\neq a\in S$. As ...
user469053's user avatar
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1 vote

Finding the extreme values of $f(x,y,z)=x^2+y^2+z^2$ constraint to $x^2+y^2+z^2+xy=12$

You can avoid any complicated inequalities or Lagrange multipliers by diagonalizing the constraint: $g(x,y,z) = x^2+xy+y^2+z^2$ can be written as $\frac{1}{4}(x-y)^2+\frac{3}{4}(x+y)^2+z^2$. Thus ...
krm2233's user avatar
  • 4,558

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