The graph of the marginal cost function from the production of x thousand bottle

Question

The graph of the marginal cost function from the production of x thousand bottles of sunscreen per month [where cost C(x) is in thousands of dollars per month] is given in the figure on the right. Complete parts (a) through (d) below. C'(x) 70- 35- 0- er month where is he cost un e on C (A Using the graph shown, describe the shape of the graph of the cost function C x as x ?ncreases from 0 to 6,000 bottes increasing and decreasing? O A. C(x) is increasing fromx-0 tox 6. ? and is increasing from x#3 tox-6 B. C(x) is decreasing from x-otox 3 O c, c(x) is increasing from x = 0 to x = 3 and is decreasing from x = 3 to x-6 O D. C(x) is decreasing from x = 0 to x-6 Describe the concavity of C(x). Choose the correct answer below. OA. C(x) is concave downward from x 0 to x 3 and is concave upward from x 3 to x 6. O B. C(x) is concave upward from x = 0 to x-3 and is concave downward from x-3 to x-6. O c, c(x) is concave downward from x = 0 to x = 6. ( D. C(x) is concave upward from x-0 to x-6.

Explanation / Answer

A)Marginal cost it the derivative of total cost.
Since marginal cost is always positive, it means that total cost is always increasing as a positive derivative means the function is increasing. Hence C(x) is increasing from x=0 to x=6

Next question:
Function will be concave up if its double derivative is >0 or derivative is increasing
Function will be concave down it its double derivative is <0 or derivative is decreasing
So, since C'(x) is decreasing from x=0 to x=3 we can say that C(x) is concave down from x=0 to x=3
Similarly, C'(x) is increasing from x=3 to x=6 we can say taht C(x) is concave up from x=3 to x=6
Hence answer is option A

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