# Tagged Questions

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### Proving the AM-GM Inequality with Lagrange Multipliers

Exercise: Let $x_1,x_2,...,x_n$ be real positive numbers. Prove the arithmetic-geometric mean inequality, $(x_1x_2...x_n)^{1/n}\le (x_1+x_2+...+x_n)/n$. Hint: Consider the function ...
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### How can I find the Min and max of this question?

I have been trying for the past 2 hours on this question and cannot seem to figure out the answer. So far I have gotten the 'green' bits correct. Someone Help please
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### How can I solve this using the Lagrange method?

This is what I keep doing but the answer seems to be wrong every time.
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### Maximum and minimum distance from the origin

Find the maximum and minimum distances from the origin to the curve $5x^3+6xy+5y^2-8=0$ My attempt: We have to maximise and minimise the following function $x^2+y^2$ with the constraint that ...
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### How can I calculate the maximum and minimum of this function?

This is incorrect and I have a feeling I am not doing this correctly.
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### How do I find the highest and lowest points made by the union of these two functions using Lagrange Multipliers?

Find the highest and lowest points made by the union of these two functions using Lagrange Multipliers. $x^2+y^2+z^2 = 16$ $(x+1)^2+(y+1)^2+(z+1)^2 = 27$ I got the basics down, I used the first ...
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### What's wrong with this Kuhn-Tucker optimization?

The function $u(x,y,z) = xyz$ is to be maximized, under constraints: $0 \le x \le 1, y \ge 2, z \ge 0$ and $4 - x - y - z \ge 0$ Now I'm not quite sure how to translate the x-constraint into ...
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### Finding constrained optima of $f(x,y) = x^3 - y^3 -x$

For a function $$f(x,y) = x^3 - y^3 -x$$, the minimum and maximum under the constraint $$x^2 + y^2 =1$$ is searched. So as usual, my approach is to set up the Lagrangian and the FOC:  L(x,y, ...
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### Minimize a three variable function using Euler-Lagrange theorem

I have to minimize the function $g(x,y,z)=x^2+y^2+2z^2-x-yz$ in two cases. First, with the restriction $x+y+z=35$, and after with $x+y+z\geq35$ I know how to do this using Lagrange multipliers ...
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### Trouble with geometrical application of Lagrange multiplier

Let $A\in\mathbb R^{n\times n}$ be positive-definite and $\langle Ax,x\rangle=1$ be the equation of an ellipsoid $M\subset\mathbb R^n$. Use Lagrange multipliers to prove that the greatest distance of ...
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### Deriving demand functions given utility

A consumer purchases food $X$ and clothing $Y$. Her utility function is given by: $U(X,Y) = XY +10Y$, income is $\$100$the price of food is$\$1$ and the price of clothing is $P_y$. Derive the ...
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### Lagrange multipliers for finding geodesics on a sphere

The problem statement, all variables and given/known data Find the geodesics on a sphere $g(x,y,z)=x^{2}+y^{2}+z^{2}−1=0$ arc-length element $ds=\sqrt{dx^{2}+dy^{2}+dz^{2}}$ Relevant equations ...
### Find an equation of the plane that passes through the point $(1,2,3)$, and cuts off the smallest volume in the first octant. *help needed please*
Find an equation of the plane that passes through the point $(1,2,3)$, and cuts off the smallest volume in the first octant. This is what i've done so far.... Let $a,b,c$ be some points that the ...
Use Lagrange multiplier to find absolute maximum and minimum of $f(x,y) =x^2+xy+y^2, x^2+y^2 =8$. What i've done so far.. \$f_x = \lambda g_x \Rightarrow 2x+y =\lambda2x, \\f_y = \lambda g_y ...