Kay's Math: Lagrange Multipliers

1. At what point or points on the circle $x^{2}+y^{2}=1$ does f(x,y)=xy have an absolute maximum, and what is that maximum?

2. Use Lagrange multipliers to find the minimum and maximum values of f subject to the given constraints. Also find the points at which these extreme values occur.

$\\1\cdot\;f(x,y)=xy;4x^{2}+8y^{2}=16 \\ \; \;2\cdot\;f(x,y)=x^{2}-y^{2}; \: x^{2}+y^{2}=25 \\ \;3\cdot\;f(x,y)=4x^{3}+y^{2};2x^{2}+y^{2}=1\\ \;4\cdot\;f(x,y)=x-3y-1;x^{2}+3y^{2}=16\\ \;5\cdot\;f(x,y)=2x+y-2;x^{2}+y^{2}+z^{2}=4\\ \;6\cdot\;f(x,y,z)=3x+6y+2z;2x^{2}+4y^{2}z^{2}=70\\ \;7\cdot\;f(x,y,z)=xyz;x^{2}+y^{2}+z^{2}=1\\ \;8\cdot\;f(x,y,z)=x^{4}+y^{4}+z^{4};x^{2}+y^{2}+z^{2}=1$

3. Suppose that the temperature at a point (x,y) on a metal plate is $T(x,y)=4x^{2}-4xy+y^{2}$ . An ant walking on the plate traverses a circle of radius 5 centered at the origin. What are the highest and lowest temperatures encountered by the ant?

4. Find all relative extrema of $x^{2}y^{2}$ subject to the constraint $4x^{2}+y^{2}=8$

Stationary points-http://ellipticalkoyal.blogspot.com/2019/01/stationary-points-of-two-variable_14.html

Kay's Math

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Wednesday, January 16, 2019

Lagrange Multipliers

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