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Problem 100

# As a result of increasing energy costs, the growth rate of the profit of the 4-yr old Venice Glassblowing Company has begun to decline. Venice's management, after consulting with energy experts, decides to implement certain energy-conservation measures aimed at cutting energy bills. The general manager reports that, according to his calculations, the growth rate of Venice's profit should be on the increase again within 4 yr. If Venice's profit (in hundreds of dollars) $$t$$ yr from now is given by the function $$P(t)=t^{3}-9 t^{2}+40 t+50 \quad(0 \leq t \leq 8)$$ determine whether the general manager's forecast will be accurate. Hint: Find the inflection point of the function $$P$$ and study the concavity of $$P$$.

Expert verified
The inflection point of the profit function $$P(t)$$ occurs at $$t=3$$. Before the inflection point, the function is concave down, indicating a decrease in profit growth rate. After the inflection point, the function is concave up, indicating an increase in profit growth rate. Therefore, the general manager's forecast is accurate, and the growth rate of Venice's profit will increase again within 4 years.
See the step by step solution

## Step 1: Find the first and second derivative of the function

Let's find the first and second derivative of the given profit function $$P(t)$$. The first derivative represents the rate of change of the profit function, i.e., the profit growth rate: $$P'(t) = \frac{dP}{dt} = 3t^2 - 18t + 40$$ Now, let's find the second derivative, which will help us analyze the concavity of the function: $$P''(t) = \frac{d^2P}{dt^2} = 6t - 18$$

## Step 2: Find the inflection point

To find the inflection point, we need to find when the second derivative equals zero: $$P''(t) = 6t - 18 = 0$$ Solving for $$t$$: $$t = \frac{18}{6} = 3$$ The inflection point occurs at $$t = 3$$.

## Step 3: Analyze the concavity of the function

To analyze the concavity of the function, we need to check the sign of the second derivative before and after the inflection point $$t = 3$$. 1. Choose a value of $$t < 3$$, say $$t = 2$$: $$P''(2) = 6(2) - 18 = -6$$ Since $$P''(2) < 0$$, the function is concave down (i.e., there's a decrease in the profit growth rate) before the inflection point. 2. Choose a value of $$t > 3$$, say $$t = 4$$: $$P''(4) = 6(4) - 18 = 6$$ Since $$P''(4) > 0$$, the function is concave up (i.e., there's an increase in the profit growth rate) after the inflection point. Since the concavity of the function changes from negative to positive around the inflection point, we can conclude that the general manager's forecast is accurate. The growth rate of Venice's profit will indeed increase again within 4 years.

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