Math 1520 Course Notes Sections: 6.1, 6.2
Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema Math 1520 Course Notes Sections: 6.1, 6.2 Liliana Menjivar Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema Section 6.1: Absolute Extrema absolute max or min extreme value theorem closed interval method first derivative test for absolute extreme values second derivative test for absolute extreme values Section 6.2: Applications of Extrema guidelines applications Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema absolute max or min The largest possible value of a function is called the absolute maximum. The smallest possible value of a function is called the absolute minimum. Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema absolute max or min Definition (Global Max/Min) Suppose f (x) is defined on some interval I . Let c be a number in I . Then I f (c) is the absolute (or global) maximum of f on I if f (c) ≥ f (x), I ∀x in I . f (c) is the absolute minimum of f on I if f (c) ≤ f (x), ∀x in I . A function has a absolute extremum (plural: extrema) of f (c) if it has either an absolute max or absolute min there. Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema extreme value theorem Theorem (Extreme Value Theorem(EVT)) If function f is continuous on a closed interval [a, b], then f attains both an absolute maximum and an absolute minimum on [a, b]. The EVT guarantees the existence of absolute extrema for a continuous function defined on a closed interval. Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema extreme value theorem Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.2: Applications of Extrema Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema extreme value theorem What if the hypotheses of the EVT fail? That is, what if f is discontinuous? what if f is not defined on a closed interval Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema extreme value theorem Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.2: Applications of Extrema Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema closed interval method To find absolute extrema for a continuous function f on [a, b] we use the closed interval method (CIM). 1. Find all critical numbers of f in (a, b) . 2. Evaluate f at its critical number(s) and at the endpoints a and b. That is, find f (c), f (a), and f (b). 3. The absolute maximum of f on [a, b] is the largest value found in step 2, and the absolute minimum of f on [a, b] is the smallest value found in step 2. Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema closed interval method Example 1. Find the absolute maximum and minimum of f (x) = x 3 − 3x 2 + 1, Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 1 [− , 4]. 2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema closed interval method Example 2. Find the absolute maximum and minimum of the average cost for the given cost function: C (x) = Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 x 2 + 36 , 2 [1, 12]. Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema closed interval method Example 3. From information given in a recent business publication, a mathematical model was constructed to represent the miles per gallon used by a certain car at a speed of x mph: 1 2 x + 2x − 20, 30 ≤ x ≤ 65. 45 Find the absolute maximum miles per gallon and the absolute minimum miles per gallon and the speeds at which they occur. M(x) = − Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema first derivative test for absolute extreme values Theorem (The First Derivative Test for Absolute Extreme Values) Suppose f is continuous on an interval I and that f has exactly one critical number x = c in I , 0 0 0 0 I If f (x) > 0 on ∀x < c and f (x) < 0 ∀x > c, then f (c) is the absolute maximum of f . I If f (x) < 0 on ∀x < c and f (x) > 0 ∀x > c, then f (c) is the absolute minimum of f . Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema second derivative test for absolute extreme values Theorem (The Second Derivative Test for Absolute Extreme Values) Suppose f is continuous on an interval I and that f has exactly one critical number x = c in I , 00 I If f (x) < 0 ∀x ∈ I , then f (c) is the absolute maximum of f . I If f (x) > 0 ∀x ∈ I , then f (c) is the absolute minimum of f . 00 Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema guidelines (1) Understand the problem. (a) Determine what is given and what is unknown. (b) Determine the quantity that is being maximized or minimized. (2) Draw a diagram. (3) Introduce notation. (a) Create variables for all the known and unknown quantities. (b) Create a variable for the quantity being maximized or minimized (for now call it Q). Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema guidelines (4) Find any relationships between the variables in step 3. (5) Express Q as a function of all other variables and constant. Then, using step 4 express Q as a function of only one variable and state the domain of Q. Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema guidelines (6) Find the absolute maximum or minimum values of Q. (a) If f continuous and D = [a, b] use CIM. (b) If f has one critical number and D = I use the first derivative test for absolute extreme values. (c) If f has one critical number and D = I use the second derivative test for absolute extreme values. (d) If f has more than one CN and D = I , find any local extrema and then calculate the limit of f at the endpoints of D to determine if the local extrema are absolute extrema. Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema applications Example 1. A farmer has 500 meters of fencing with which to fence in three sides of a rectangular pasture. A straight river will form the fourth side. Find the dimensions of the pasture of the largest area that the farmer can fence. Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema applications Example 2. A theater owner charges $5.00 per ticket and sells 250 tickets. By checking other theaters, the owner decides that for every one dollar he raises the ticket price, he will lose 10 customers. What should he charge to maximize revenue? Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema applications Example 3. A company wants to manufacture cylindrical aluminum cans that hold 1000 cm3 (1 liter) of oil. What should the radius and height of the can be to minimize the amount of aluminum used? Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema applications Example 4. A carpenter wants to make an open-topped box out of a square piece of tin, 12 in wide, by cutting out a square from each of the four corners and bending up the sides. What size square should be cut from each corner to produce a box of maximum volume? Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2 Section 6.1: Absolute Extrema Section 6.2: Applications of Extrema applications Example 5. A company manufactures and sells x smart phones per week. The weekly demand and cost equations are, respectively, p = 500 − .5x, C (x) = 20, 000 + 135x (a) What price should the company charge for the phones, and how many phones should be produced to maximize the weekly revenue? What is the maximum weekly revenue? (b) What is maximum weekly profit? How much should the company charge for the phones, and any phones should be produced to realize the maximum weekly profit? Liliana Menjivar Math 1520 Course Notes Sections: 6.1, 6.2
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