Use the method of Lagrange multipliers to find among all scalar potential functions for F one,

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4. Let F be the vector field 

F = (6x2 — 6xy, 3y2 3x2)

and let C be the circle x2 ± y2 = 2 of radius 0 centred at (0,0).

(a) The vector field F is conservative. Find a scalar potential function for F.

(b) Use the method of Lagrange multipliers to find among all scalar potential functions for F one, ¢0, whose minimum on the circle C is 0. What is the maximum of 00 on C? (

c) If 00 is scalar potential function for F from (b) evaluate the line integral fc 0,3 ds of 00 along C. 

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