(a) Let S be the set in R^2 de%uFB01ned by these simultaneous linear inequalitie

Question

(a) Let S be the set in R^2 de%uFB01ned by these simultaneous linear inequalities:

x1 %u2265 0, x2 %u2265 0, x1 %u2212%u221A3 x2 %u2265 %u22122%u221A3, x1 +%u221A3 x2 %u2265%u221A3, x1 + x2 %u2264 3%u221A3.

Make a careful sketch of S.

(b) Find the coordinates of all the corner points of S.

(c) Find the exact interior angle (in radians) at each corner point of S. Any logically correct

method is acceptable.

(d) Find the point of S that gives the largest possible value to %u03B6(x1, x2) = x1 %u2212 x2. Label this

point A on your sketch from part (a); include its exact coordinates on the %uFB01gure. Working

counterclockwise around S, label the other corner points B, C, . . .; give their coordinates, too.

(e) For each nonzero vector c = (c1, c2), %uFB01nd all points in S that maximize the function %u03B6(x) = c^Tx.

Present your results both algebraically and graphically, using a %u201Cmap%u201D drawn on the (c1, c2)-

plane.

(f) Let Z = Z(c) denote the maximum value of c^Tx when x %u2208 S, expressed as a function of the

vector c. In each region of the (c1, c2)-plane described in part (e), there is a simple formula for

Z(c). Find all such formulas and the regions where they apply

Explanation / Answer

x1 %u2265 0, x2 %u2265 0, x1 %u2212%u221A3 x2 %u2265 %u22122%u221A3, x1 +%u221A3 x2 %u2265%u221A3, x1 + x2 %u2264 3%u221A3. what is it i cant understand

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