Question

Use Lagrange multiplier method to find the maximum and minimum values of (, , ) = − 2 + 5 On the sphere 2 + 2 + 2 = 30

Answer 1

Please find the complete question in the attached file.

Explanation:

Calculation with Lagrange multipliers of its optimum restricted places,

`f_x = 2\ \ \ \ \ \ \ \ \ \ \ \ \ g_x = 2x\\\\f_y = 1 \ \ \ \ \ \ \ \ \ \ \ \ \ g_y = 2y\\\\`

Set the multiplier formulas for Lagrange:

`f_x = \lambda g_x \to 2 = \lambda 2x ............ (i)\\\\f-y = \lambda g_y \to 1 = \lambda 2y................ (ii)\\\\constraint: \\\\\to x^2 + y^2 = 5 .................... (iii)\\\\Taking \ (i)\ divides\ (ii), \ (assuming\ \lambda \\eq 0)\\\\(2)/(1)=(\lambda 2x)/(\lambda 2y)=(x)/(y)\\\\\therefore \\\\2y = x\\\\Sub\ into\ (iii)\ to\ find\\\\4y^2 + y^2 = 5 \to y = \pm 1\\\\`

We get solutions (x,y)=(2,1) and (−2,−1) if combined with 2y = x. These are all the identical points we obtained in (c) and we know the f(2,1) is the max, whereas f(−2, −1) is a minimum.