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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.

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Chapter 10

The average revenue is defined as the function $$ \bar{R}(x)=\frac{R(x)}{x} \quad(x>0) $$ Prove that if a revenue function \(R(x)\) is concave downward $\left[R^{\prime \prime}(x)<0\right]$, then the level of sales that will result in the largest average revenue occurs when \(\bar{R}(x)=R^{\prime}(x)\).

Chapter 10

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=\frac{1}{1+x^{2}} $$

Chapter 10

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=x^{2}-2 x-3 \text { on }[-2,3] $$

Chapter 10

Find the absolute maximum value and the absolute minimum value, if any, of each function. $$ f(x)=x e^{-x^{2}} \text { on }[0,2] $$

Chapter 10

The total world population is forecast to be \(P(t)=0.00074 t^{3}-0.0704 t^{2}+0.89 t+6.04 \quad(0 \leq t \leq 10)\) in year \(t\), where \(t\) is measured in decades with \(t=0\) corresponding to 2000 and \(P(t)\) is measured in billions. a. Show that the world population is forecast to peak around 2071 . Hint: Use the quadratic formula. b. What will the population peak at?

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