A manufacturer produces three products; Product X, Product Y and Product Z. To be able to produce each product he uses a combination of three distinct inputs, namely, Input I, Input II and Input III, the supply of which is currently at 300, 545 and 430 units respectively. Each unit of Product X requires 2 units of Input I, 5 units of Input II and 1 unit of Input III to be produced. Similarly, each unit of Product Y requires 3 units of Input 1 and 3 units of Input II while Product Z uses 1 unit of Input I, 2 units of Input II and 4 units of Input III.
a. Let x, y and z represent the number of units produced of Products X, Y and Z respectively. If the manufacturer must decide what quantity of each product to produce so that his entire supply of inputs is utilized, formulate his or her decision in the form of a system of three linear equations in the variables x, y and z.
b. Rewrite the system of linear equations in the matrix form.
c. Solve the system of linear equations using the Inverse method.
2. A study found that average driving costs of sports car ( measured in cents per km), is approximated by the function C(x) = 4040 + 28.50, where x denotes the number______of kilometres (in thousands), that the car is driven in a year.
i. What is the average cost of driving the car when 7000 km were driven in a year?
ii. What is the average cost of driving the car when 23000 km were driven in a year?
iii. What happens to the average cost as the number of kilometres driven increases without bound?
7 if 0 ≤ x < 2
Is f(x) = {x2 + 3 if 2 ≤ x < 4
6x - 5 if x ≥ 4
a. Determine whether f(x) continuous everywhere throughout its domain?
4. Consider the function f(x) = 1x3 + 2x2 - 21x + 15 = 3
a. For what value(s) of x is the function concave?
b. Find and classify the stationary points on the function f(x)