Activity 2: Determine the absolute extrema of the function f(x) = x - 12x on [0, 4]. Show complete solution using the four steps. Step 1. Determine if the polynomial is continuous. Step 2. Find the first derivative and find the critical points. Step 3. Evaluate the function at the critical points and the endpoints of the interval. Note only critical value that lie in the given interval. Step 4. Determine the maximum point (highest value of f() and the minimum point (lowest value of f(x), if there is any. Analysis: Express what you have learned in these lessons/ activities by answering the questions below. 1. When can we say that a function or a graph has extrema (both minimum and maximum value)? 2. How do we identify the minimum and maximum value of the graph of a function at a given interval? 4. How can we solve optimization problem? Assessment: Multiple Choice. Read and analyze each question. If answer is not among the choices, write the correct one. 1. What is the first thing to consider when finding the extrema of a function? A. Check the critical points. C. Identify the asymptotes. B. Evaluate x on the given interval. D. Verify whether continuous or not. 2. In what interval does x=4 lie? A. [-2, 4) B. (0, 3] C. [-2, 6] D. [-1, 2] For numbers 3 and 4, study the graph at the right using the interval [-5, 5] 3. What is the maximum point? A. (-3, 2) B. (-1, -3) C. (2, 1) D. (4, - 1) 4. At what value of x is fminimum? A. 4 B. 2 C. -1 D. -3 For numbers 5 and 6, use f(x) = -x3- 14x2-60x - 75on the interval [-7. -5]. 5. Which is the absolute minimum? A. (-7,-6) B. (-6, -3) C. (-5,-1) D. (-4, 0) 6. Which is the absolute maximum? A. (-7,2) B. (-7, 0) C. (-6,3) D. (-5,4)7. Which of the following can be candidates for absolute extrema? I. asymptotes of the function Ill. endpoints of the closed interval II. critical points of the function IV. limits of the given function A. I and II only B. I and IV only C. II and III only D. III and IV only For numbers 8 to 11: An open-typed rectangular box is to be made from a piece of cardboard 24 cm long and 9 cm wide by cutting out identical squares from the four corners and turning up the sides. 8. Which of the following illustrates the volume of the box? A. V(h) = h(24 - h)(9 -h) C. V(h) = h(9 - 2h)(24-2h) B. V(h) =h(24+ h)(9+h) D. V(h)= h(9+ 2h)(24 -2h) 9. What is the first derivative of V(h)? A. V(h) = 240 - 124h + 4h2 C. V(h) = 140 - 132h + 12h2 B. V(h) = h(24 + h) (9 + h) D. V(h) = 216 - 132h + 12h2 10. Which could be the height of the box? A. 1.75 cm B. 2.00 cm C. 2.10 cm D. 2.50 cm 1 1. What could be the maximum volume of the formed box? A. 210 cm B. 200 cm C. 190 cm D. 180 cm3 For numbers 12 to 13: A farmer has 2,400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. 12.Which of the following gives the derivative of A(x)? A. 2, 400 - x B. 2, 400 - 4x C. 1,200-x D. 1,200 - 2x 13.What are the dimensions of the field that can give the maximum area? A. 500 ft x 1,400 ft C. 700ftx 1,000fc B. 600 fx 1,200 ft D. 800ft x 800 ft For number 14 to 15: A manufacturer needs to make a cylindrical can that will hold 2 liters of liquid. Determine the dimensions of the can that will minimize the amount of material used in its construction. 14. What is the approximate measure of the radius of the cylindrical can made? A. 5.78 cm B. 6.29 cm C. 6.83 cm D. 7.18 cm 15.Which is the approximated height of the cylindrical can? A. 11.56 cm B. 12. 58 cm C. 13.66 cm D. 14.36 cm Adapted and modified from the SLM of Department of Education - Region illActivity 1. What's My Extrema? Identify the extrema (both minimum and maximum) of the given graphs below. If there are no extrema, provide an explanation. 4. 1. 1 09 72 2 2 5. 2. 2 3