Absolute Extrema and Optimization

A. Find the absolute maximum value and the absolute minimum value, if any, of the function.

1. \(f(x)=3-4x+x^2\)   solution

2. \(g(x)=-x^2+2x+6\)  solution

B. Find the absolute maximum and absolute minimum values of \( f \) on the given interval.

1. \(f(x)=-x^2+4x+1, \quad [4,7] \)    solution

2. \(f(x)=3x^2-12x+5, \quad [0, 3] \)    solution

3. \(f(x)=x^3-6x^2+9x+2, \quad [-2, 2]\)    solution

4. \( f(x) = x + \dfrac{1}{x}, \quad [0.5, 3] \)    solution

5. \(f(t)=\dfrac{t}{t-1}, \quad [3, 5] \)  solution

C. An apartment complex has 100 two-bedroom units. The monthly profit (in dollars) realized from renting out \(x\) apartment units is given by the following function \[ P(x)=-10 x^{2}+1,620 x-42,000. \] To maximize the monthly rental profit, how many units should be rented out? What is the maximum profit realizable?  solution

D. A small company spends up to $8,000 on advertising. The company’s profit (in thousands of dollars) depends on the amount of money \(x\) (in thousands of dollars) they spent on advertising each month according to the rule \[P(x) = −x^2 + 8x + 20.\] What should the company’s monthly advertising be to maximize monthly profits? What is the company’s maximum monthly profit?  solution

E. A production facility is capable of producing 80,000 widgets in a day and the total cost of producing \(x\) widgets in a day is given by \[ \displaystyle{C(x)=0.67x+250,000+\frac{16,750,000}{x}}.\] How many widgets per day should they produce in order to minimize production cost? What is the minimum production cost?   solution

F. The quantity demanded each month of the Walter Serkin recording of Beethoven's Moonlight Sonata, produced by Phonola Media, is related to the price per compact disc. The equation \[ p=-0.00048 x+10 \quad(0 \leq x \leq 12,000) \] where \(p\) denotes the unit price in dollars and \(x\) is the number of discs demanded, relates the demand to the price. The total monthly cost (in dollars) for pressing and packaging \(x\) copies of this classical recording is given by \[ C(x)=600+2 x-0.00002 x^{2} \quad(0 \leq x \leq 20,000) \] To maximize its profits, how many copies should Phonola produce each month?   solution