the monthly output of a product (in units) is given by ( p = 1300x^{\frac{5}{2}} ), where ( x ) is the…

Explanation:

Step1: Analyze the function (P = 1300x^{\frac{5}{2}})

The function (P = 1300x^{\frac{5}{2}}=1300\sqrt{x^{5}}) is a power - function. When (x = 0), (P=0). As (x) increases from (0) to (10), we can calculate some values: When (x = 1), (P = 1300\times1^{\frac{5}{2}}=1300); when (x = 4), (P=1300\times4^{\frac{5}{2}}=1300\times32 = 41600); when (x = 9), (P=1300\times9^{\frac{5}{2}}=1300\times243=315900). The first - derivative of (y = ax^{n}) is (y^\prime=anx^{n - 1}). For (P = 1300x^{\frac{5}{2}}), (P^\prime=1300\times\frac{5}{2}x^{\frac{3}{2}}=3250x^{\frac{3}{2}}). The second - derivative (P^{\prime\prime}=3250\times\frac{3}{2}x^{\frac{1}{2}}=\frac{9750}{2}\sqrt{x}).

Step2: Determine the concavity

Since (x\in[0,10]) and (P^{\prime\prime}=\frac{9750}{2}\sqrt{x}>0) for (x\in(0,10]) and (P^{\prime\prime}(0) = 0) (the function is smooth at (x = 0)), the function (P = 1300x^{\frac{5}{2}}) is concave up. For the graph, as (x) increases, the rate of increase of (P) (since (P^{\prime\prime}>0)) is increasing.

Answer:

a. The correct graph is A. b. The graph is concave up.