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Problem 156

Find the values of \((x, y, z)\) that minimize $$ F(x, y, z)=x y+2 y z+2 x z $$ given the condition \(\mathrm{G}(\mathrm{x}, \mathrm{y}, z)=\mathrm{xyz}=32\).

Expert verified

The values of \((x, y, z)\) that minimize the function \(F(x, y, z)=xy+2yz+2xz\) given the constraint \(G(x, y, z)=xyz=32\) are \((x, y, z) = (4, 4, 2)\).

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Chapter 6

a) Find whether the origin is a relative maximum or minimum, or neither for the function $\mathrm{f}(\mathrm{x}, \mathrm{y})=\log \left(1+\mathrm{x}^{2}+\mathrm{y}^{2}\right)$. b) Find the critical points of the function $$ f(x, y)=x^{2}-12 y^{2}+4 y^{3}+3 y^{4} $$ and determine whether each is a relative maximum or saddle point of \(\mathrm{f}(\mathrm{x}, \mathrm{y})\)

Chapter 6

\(\mathrm{M}\) where \(\mathrm{p}_{1}=1, \mathrm{p}_{2}=2\) and \(\mathrm{M}=10 .\) Check the second-order conditions to verify that the solution is indeed a maximum.

Chapter 6

Find the critical points of the function $$ f(x, y)=x^{4}+y^{4}-x^{2}-y^{2}+1 $$ Then determine if each critical point is a relative maximum, relative minimum, or saddle point.

Chapter 6

Find the point of the plane \(2 \mathrm{x}-3 \mathrm{y}-4 \mathrm{z}=25\) which is nearest the point \((3,2,1)\).

Chapter 6

For the following quadratic forms, tell by inspection whether the origin is a maximum or minimum: \(\mathrm{q}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2}+\mathrm{y}^{2}\) \(q(x, y)=x^{2}-y^{2}\) \(q(x, y)=x y\)

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