The formula t=P−1−−−−−√3+1

Question

The formula  t=P−1−−−−−√3+1
  can be used to predict the number of years it will take a company to have a profit of P dollars, in ten thousands.

Where is the inflection point located and what does it indicate?

Select the correct answers from the lists.

(1 point)
The inflection point is located at 
.

It represents that the profit of the company will be approximately 
 after 1 year.

GPT-4o mini · Answer

To analyze the formula $ t = \sqrt{P - 1} \cdot (3 + 1) $, we first need to understand what an inflection point is. An inflection point occurs at a position on the curve where the curvature changes, which means that the rate of growth of the function changes from increasing to decreasing or vice versa. To find the inflection point we typically need to take the second derivative of the function with respect to $ P $ and set it to zero. The information provided is limited, but if we assume you want to find the point for that specific formula, we would first rewrite the formula in a more simplified way. From the function $ t = \sqrt{P - 1} \cdot 4 $, 1. If we differentiate, we can find $ \frac{dt}{dP} $ (the first derivative). 2. Then, we’d differentiate again to find the second derivative $ \frac{d^2t}{dP^2} $. Let's apply this method step by step: 1. **Calculate the first derivative $ \frac{dt}{dP} $:** $$ \frac{dt}{dP} = \frac{1}{2\sqrt{P - 1}} \cdot 4 = \frac{2}{\sqrt{P - 1}} $$ 2. **Calculate the second derivative $ \frac{d^2t}{dP^2} $:** $$ \frac{d^2t}{dP^2} = -\frac{2}{2(P - 1)^{3/2}} = -\frac{1}{(P - 1)^{3/2}} $$ This second derivative is negative for all $ P > 1 $, indicating that there is no change in curvature and thus no inflection point in terms of profit $ P $. However, if you meant to derive this conceptually, note that a function $ t = \sqrt{P - 1} $ grows as $ P $ increases, but the growth rate (the first derivative) is decreasing as $ P $ increases. Considering this, without a specific numerical threshold supplied for profit $ P $, we are limited in giving an exact location for the inflection point. So, if we were to answer your questions based on the understanding that often the company starts seeing profit just above $ P = 1 $: ### Answer Selection 1. The inflection point is located at **1** (if $ P = 1 $ is fetchable from practical interpretation). 2. It represents that the profit of the company will be approximately **$10,000** (1 in 10,000). Let me know if you require further analysis or a different context for the inquiry!