The point (4,3.5) is the maximum of the profit function. A. Explain what the point (4,3.5) represents in terms of the context of the graph.

The function shown describes f(x), the cost (in hundreds of thousands of dollars) of making x bicycles (in thousands).

F(x)=1.25+0.35x

B. what is the cost to make 4,000 bicycles? Show your work.

Compare the slope of the cost function with the graph of the profit function.

C. Explain why the company should plan on making fewer than 4,000 bicycles.

A. The point represents the turning point of the graph

B. The cost of making 4,000 bicycles is $265,000

The slope of the cost function, F(x) = 1.25 + 0.35·x, is less than the average slope of the graph of the total profit for sales

C. The company should plan on making fewer than 4,000 bicycles, to maximize profit

Step-by-step explanation:

A. The point represents the turning point after which increase in the x-coordinate values leads to a decrease in the y-coordinate values

B. The cost function is presented as follows;

F(x) = 1.25 + 0.35·x

Where;

F(x) = The cost in hundreds of thousands of dollars of making 'x' (in thousands) bicycles

x = The number of bicycles made in thousands

The cost of making 4,000 bicycles is given by plugging in x = 4(thousand) in  F(x) = 1.25 + 0.35·x, as follows;

F(x) = 1.25 + 0.35·x

∴ F(4) = 1.25 + 0.35 × 4 = 2.65

Therefore, the cost of making 4,000 bicycles = 2.65 × $100,000 = $265,000

The slope of the cost function, F(x) = 1.25 + 0.35·x, is given by the coefficient of 'x' which is 0.35

The average slope of the graph of the profit function is ((3.5 - 0)/(4 - 0)) = 0.875

The slope of the cost function is less than the average slope of the graph of the profit function taken from the origin to the maximum point of the graph

C. When the company produces 5,500 bicycles, x = 5.5, we have;

F(5.5) = 1.25 + 0.35 × 5.5 = 3.175

The cost is 3.75 × $100,000 = $375,000 and the profit from te graph is $300,000

Therefore, the cost of making the 5,500 bicycles is more than the profit made in sales

The company should plan on making fewer than 4,000 bicycles, to reduce cost of producing more bicycles and to increase profit per unit of bicycle sold