minimum using Lagrange multipliers minimum using Lagrange multipliers I need to

Question

minimum using Lagrange multipliers minimum using Lagrange multipliers I need to find the minimum x,y, and z coordinates of the paraboloid x^2+y^2=z with the constraint 6x-6y+108=36z. Please show me all the steps

Explanation / Answer

Let F(x,y,z) = x²y²z² and G(x,y,z) = x² + y² + z² = 1. Then, ?F(x,y,z) = ? ?G(x,y,z) ==> (2xy²z², 2x²yz², 2x²y²z) = ?(2x, 2y, 2z) Thus, 2xy²z² = ?(2x) ==> x(y²z² - ?) = 0 2x²yz² = ?(2y) ==> y(x²z² - ?) = 0 2x²y²z = ?(2z) ==> z(x²y² - ?) = 0. If x = 0 (and ? is nonzero), then the other equations imply that y = z = 0. This is a contradiction, because x² + y² + z² = 1. (Likewise for the other variables.) If ? = 0, then at most two of x,y,z = 0 (due to the constraint equation). All such critical values yield F(x,y,z) = 0. So, we assume that x,y,z are all nonzero. ==> ? = y²z² = x²z² = x²y². ==> x²y²z² = ?x² = ?z² = ?y². Adding, we see that 3x²y²z² = ?x² + ?z² + ?y² = ?(x² + z² + y²) = ?. Finally, 3x²y²z² = ? = y²z² ==> 3x² = 1 ==> x = (+,-) 1/sqrt(3). Similarly, y = z = (+,-) 1/sqrt(3). This yields 8 critical points, all of which yields a value of (1/3) * (1/3) * (1/3) = 1/27. Thus, the maximum value is 1/27 and the minimum value is 0.

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