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Convex and Compact Sets and the Implicit Function Theorem

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5. Consider the set S = {(x_1, x_2) belongs to R^2: x_1 + x_2 = 1 and x_1 > 0, x_2 >= 0}.Prove that S is convex and compact.

6. Consider the function: F(x_1, x_2, y) = (x_1)^2 - (x_2)^2 + y^3.

a) Show that the equation F(x_1, x_2, y) = 0 defines y as an implicit function of x_1 and x_2 near the point (x_1, x_2) = (6,3).

b) Use the implicit function theorem to compute: dy/dx_1 and dy/dx_2 at the point (x_1, x_2) = (6, 3).

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This solution provides the proof for the convex and compact sets in an attached Word document.

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