Use Lagrange multipliers to find the points on the given surface y^(2)=16+xz that are closest to the origin. {:[" (smaller "y"-value) ",(x","y","z)=(◻)],[" (larger "y"-value) ",(x","y","z)=(◻)]:}

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Define Objective Function: Define the objective function and the constraint. The objective function to minimize is the distance squared from the origin, which is D(x, y, z) = x^2 + y^2 + z^2. We square the distance to avoid dealing with square roots, which simplifies the calculus without affecting the location of the minimum. The constraint is given by the equation of the surface, G(x, y, z) = y^2 - 16 - xz = 0.Set Up Equations: Set up the system of equations using Lagrange multipliers. To find the points closest to the origin, we use the method of Lagrange multipliers. This involves setting the gradient of the objective function equal to a constant (the Lagrange multiplier, λ) times the gradient of the constraint:  ∇D(x, y, z) = λ∇G(x, y, z)  This gives us the following system of equations:  1) 2x = λ(-z)  (derivative of D with respect to x equals λ times derivative of G with respect to x) 2) 2y = λ(2y)  (derivative of D with respect to y equals λ times derivative of G with respect to y) 3) 2z = λ(-x)  (derivative of D with respect to z equals λ times derivative of G with respect to z) 4) y^2 = 16 + xz (the original constraint)Solve Equations: Solve the system of equations. From equation 1) and 3), we can see that if λ is not zero, then x = -z and z = -x, which implies x = z. Substituting x for z in the constraint equation 4) gives us y^2 = 16 + x^2. Now we can solve for x in terms of y using this equation.Find Relationship: Find the relationship between x and y. Substituting x = z into y^2 = 16 + xz, we get y^2 = 16 + x^2. Rearranging, we have x^2 = y^2 - 16. Since x = z, we also have z^2 = y^2 - 16.Analyze Data: Analyze the relationship between x, y, and z. From equation 2), we have 2y = λ(2y). If λ is not 1, then y must be 0. However, if y = 0, then the constraint equation becomes 0 = 16 + xz, which is not possible since x = z. Therefore, λ must be 1, and we do not get any additional information about y from this equation.Substitute x: Substitute x = z into the distance function. Since x = z, the distance function D(x, y, z) = x^2 + y^2 + z^2 becomes D(x, y, x) = 2x^2 + y^2. We already have x^2 = y^2 - 16 from Step 4, so we can substitute this into the distance function to get D(y) = 2(y^2 - 16) + y^2 = 3y^2 - 32.Minimize Distance: Minimize the distance function D(y) with respect to y. To find the minimum distance, we take the derivative of D(y) with respect to y and set it equal to zero:  dD/dy = 6y = 0  This implies that y = 0, which contradicts our earlier finding that y cannot be zero. This indicates a mistake has been made in the previous steps.

Use Lagrange multipliers to find the points on the given surface $y^{2}=16+x z$ that are closest to the origin. $\newline$ $\begin{array}{ll} \text { (smaller } y \text {-value) } & (x, y, z)=(\square) \\ \text { (larger } y \text {-value) } & (x, y, z)=(\square) \end{array}$

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