Answer:
Amy should produce 2 pitchers to maximize profit and the maximum profit will be $18.
Step-by-step explanation:
Amy can produce china pitchers at a cost of $3 each.
She estimates the price function that can be represented by
p = 20 - 4x
Let she sells the number of x pitchers in a week.
Then the selling price of x pitchers will be = $x(20 - 4x)
Cost price of x pitchers = $3x
Profit (P) = Selling price - Cost price
= x(20 - 4x) - 3x
= 20x - 4x²- 3x
= 17x - 4x²
To maximize the profit, ![\frac{dP}{dx}=0](https://tex.z-dn.net/?f=%5Cfrac%7BdP%7D%7Bdx%7D%3D0)
P' = 17 - 8x = 0
8x = 17
![x=\frac{17}{8}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B17%7D%7B8%7D)
x = 2.125
x ≈ 2
Therefore, 2 pitchers should be produced.
Profit = 17x - 4x²
= 17(2) - 4(2)²
= 34 - 16
= $18