Calculus Question
Calculus: Learning Worksheets Chapter 4 Name ________________________________ Date ______________ Class ____________ Section 4-5 Absolute Maxima and Minima Goal: To find absolute maxima or minima on open and closed intervals Definition: Absolute Maxima and Minima If f (c) f ( x) for all x in the domain of f, then f (c) is called the absolute maximum. If f (c) f ( x) for all x in the domain of f, then f (c) is called the absolute minimum. Theorems: 1. A function f that is continuous on a closed interval [a, b] has both an absolute maximum and an absolute minimum on that interval. 2. Absolute extrema (if they exist) must always occur at critical values or at endpoints. 3. Second Derivative Test Let f be continuous on an interval I with only one critical value c in I. If f ‘(c) 0 and f ”(c) 0, then f (c) is the absolute minimum of f on I. If f ‘(c) 0 and f ”(c) 0, then f (c) is the absolute maximum of f on I. Procedure: Finding absolute extrema on closed intervals 1. Check to make certain that f is continuous over [a, b]. 2. Find the critical values in the interval (a, b). 3. Evaluate f at the endpoints a and b and at the critical values found in step 2. 4. The absolute maximum is the largest value found in step 3. 5. The absolute minimum is the smallest value found in step 3. 151 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 1. Find the absolute maximum and the absolute minimum for the given function on the given interval. f ( x) = 7 x 2 − 14 x + 5 [−2,5] 152 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 In Problems 2–3, find the absolute maximum and the absolute minimum, if either exists, for each function. 2. 3. f ( x) 5 x3 6 x 4 f ( x) = x2 − 5 x2 + 2 153 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 In Problems 4–6, find the indicated extremum of each function on the given interval. 4. 7 Absolute minimum value on (0, ) for f ( x) 6 2 x . x 154 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 5. Chapter 4 Absolute minimum value on [0, ) for f ( x) 10 x 2 8 x 26. 155 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets 6. Chapter 4 Absolute maximum value on (0, ) for f ( x) 3 x 6 18 x 4 . 156 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 Name ________________________________ Date ______________ Class ____________ Section 4-6 Optimization Goal: To solve application problems that involve maximum and minimum values Procedure: 1. 2. 3. 4. Strategy for Solving Optimization Problems Introduce variables, look for relationships among the variables, and construct a mathematical model of the form Maximize (or minimize) f ( x) on the interval I. Find the critical values of f ( x). Use the procedure developed in Section 4-5 to find the absolute maximum (or minimum) value of f ( x) on the interval I and the value(s) of x where this occurs. Use the solution to the mathematical model to answer all the questions asked in the problem. Solve the following problems using the four-step procedure outlined above. 1. Find two numbers whose sum is 50 and whose product is maximum. 157 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 2. A fence is to be built to enclose a rectangular area of 900 square feet. The fence along three sides is to be made of material that costs $14 per foot. The material for the fourth side costs $42 per foot. Find the dimensions of the rectangle that will allow for the most economical fence to be built. 158 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 3. A commercial apple grower estimates from past records that if 50 trees are planted per acre, then each tree will yield an average of 70 pounds of apples per season. If, for each additional tree planted per acre (up to 30), the average yield is reduced by 1 pound, how many trees should be planted per acre to obtain the maximum yield per acre? What is the maximum yield? 159 Copyright © 2019 Pearson Education, Inc. Calculus: Learning Worksheets Chapter 4 4. A 400-room hotel in New York City is filled to capacity every night at $100 a room. For each $2 increase in rent, 4 fewer rooms are rented. If each rented room costs $12 to service per day, how much should the management charge for each room to maximize gross profit? What is the maximum gross profit? 160 Copyright © 2019 Pearson Education, Inc.
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