Answer: 1) The minimum price, in dollars, to avoid a loss = 22,
2) the maximum price in dollars , to avoid a loss = 53,
3) The price that results the maximum profit = 37.5 dollars
Step-by-step explanation:
Since the given function that shows the total profit,

Where x is the price of the product.
Since for avoiding the loss,
,
⇒ 
If 
If 
Thus, 53 ≥ x ≥ 22,
Therefore, the minimum price to avoid the loss = $ 22
And, the minimum price to avoid the loss = $ 53
![f'(x) =\frac{d}{dx}[(x-22)(53-x)] = (x-22)\frac{d}{dx} (53-x)+(53-x) \frac{d}{dx}(x-22)](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%5Cfrac%7Bd%7D%7Bdx%7D%5B%28x-22%29%2853-x%29%5D%20%3D%20%28x-22%29%5Cfrac%7Bd%7D%7Bdx%7D%20%2853-x%29%2B%2853-x%29%20%5Cfrac%7Bd%7D%7Bdx%7D%28x-22%29)
⇒ 
Now, For maximum or minimum,
,
⇒ 
⇒ 
⇒ 

For x = 37.5, f''(x) is negative,
Thus, For the price of $ 37.5 dollars the company has the greatest profit.