### Compute algebraically optimal range for objective function

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A nutritionist wishes her clients to have a daily minimum of 30 units of vitamin A, 20 units of vitamin D, and 24 units of vitamin E. One dietary supplement x costs \$80 per kilogram (\$80/kg) and provides 2 units of vitamin A, 5 units of vitamin D, and 2 units of vitamin E. A second dietary supplement y costs \$160/kg and provides 6 units of vitamin A, 1 unit of vitamin D, and 3 units of vitamin E.

a. Mathematically set up this minimization problem (hint: specify the objective function and constraints for vitamins A, D, and E).

b. Using the constraints, graph the feasible region for this problem. Label all lines drawn on the graph and the axes.

c. Graphically show the optimal solution for this problem (hint: using the objective function graph the iso-cost line and shift the line until it is tangent to the feasible region).

d. Use the simultaneous equations method to find the optimal solution (hint: simultaneously solve the constraints intersecting at the point of tangency).

e. Compute algebraically the optimal range for the objective function coefficient for ?? and ??. (Hint: Use the slopes of the iso-cost line and the constraints, and give the range of the objective function coefficients).

f. Compute algebraically the shadow price for the vitamin A (Hint: increase the RHS value of the constraint by 1 and find the new optimal solution, calculate the new optimal profit, and calculate the difference of the new profit and original profit).

g. What is the shadow price of vitamin D?

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