Find the local minimum, maximum or saddle points of the given function. (No cons

Question

Find the local minimum, maximum or saddle points of the given function. (No constraints.) Clearly indicate if the point found corresponds to a local min., a local max., or a saddle. f(.x, y) = 2(x - y)(1 - xy) Let z = Squareroot 2 + x^3 + y^2 and let a path in the x, y-plane be given by x = ln 2t and y = 3cos t (1/2 lessthanorequalto t lessthanorequalto pi) Find an expression for dx/dt. Express your answer in terms of t. Evaluate the line integral with the specified path. Integral_C z dx + x^3 dy + y^2 dz where C is the line segment from (2, 0, 0) to (4, 1, 3). Let F(x, y, z) = (x + y^2)i + (y + z^2)j + (z + x^2)k, and let C be the triangle whose vertices are (2, 0, 0), (0, 4, 0), and (0.0, 0). Use Stoke's Theorem to calculate line integral_C F middot dr

Explanation / Answer

F = 2(x-y)(1-xy)

F' = 2 (x-y)(0-xy'-yx') + (1-xy)(1-1)
For minima maxima F'=0

0 = -2(x-y)(xy'+yx')

x=y or x'/x=-y'/y => log x= -log y => xy=1

also, (xy'+yx')=0
xy''+y'+yx''+x'=0

F''= -2(x-y)(xy''+y'+yx''+x')

at x=y and xy=1, F'' = 0 hence (1,1) and (-1,-1) are the saddle point of the function F

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