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The Extreme Value Theorem: If Is Continuous on a Closed Interval

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The Extreme Value Theorem: If Is Continuous on a Closed Interval AP Calculus 4.1 Notes Maximum and Minimum Values Goal: Apply derivatives to calculate extreme values. Identify the maxima and minima of the following graph. Find the maxima and minima Example 1:푓(푥) = 푐표푠 푥 Example 2: 푓(푥) = 푥2 Example 3: 푓(푥) = 푥3 Example 4: 푓(푥) = 3푥4 − 16푥3 + 18푥2; 1 ≤ 푥 ≤ 4 The Extreme Value Theorem: If 푓 is continuous on a closed interval [푎, 푏], then 푓 attains an absolute maximum value 푓(푐) and an absolute minimum value 푓(푑) at some numbers of 푐 and 푑 in [푎, 푏] Examples: Non-Examples Fermat’s Theorem: If 푓 has a local maximum or minimum at c, and if 푓′(푐) exists, then 푓′(푐) = 0. Careful! When 푓′(푥) = 0 it doesn’t automatically mean there is a maximum or minimum! In other words the converse of Fermat’s Theorem is not true. Critical Numbers: A critical number of a function f is a number c in the domain of f such that either 푓′(푐) = 0 or 푓′(푐) does not exist. Example 5: Find the critical numbers 푓(푥) = 3푥2 − 2푥 푓(푥) = 푥2/3 + 1 If f has a local maximum or minimum at c then c is a critical number of f. To find an absolute maximum or minimum of a continuous function on a closed interval, we note that either it is local (in which case it occurs at a critical number) or it occurs at an endpoint of the interval. Thus the following three-step method ALWAYS works. The Closed Interval Method: To find the absolute maximum and minimum values of a continuous function f on a closed interval [푎, 푏]: 1. Find the values of f at the critical numbers of f in (푎, 푏) 2. Find the values of f at the endpoints of the interval 3. The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of these values of the absolute minimum value. 1 Example 6: Find the absolute maximum and minimum values of the function 푓(푥) = 푥3 − 3푥2 + 1; − ≤ 푥 ≤ 4 2 Example 7: The Hubble Space Telescope was deployed on April 24th, 1990 by the space shuttle Discovery. A model for the velocity of the shuttle during this mission, from liftoff 푎푡 푡 = 0 until the solid rocket boosters were jettisoned at 푡 = 126 푠푒푐표푛푑푠 is given by (in feet per second). 푣(푡) = 0.001302푡3– 0.09029푡2 + 23.61푡 – 3.083 Using this model, estimate the absolute maximum and minimum values of the acceleration of the shuttle between liftoff and the jettisoning of the boosters. AP Calculus 4.2 The Mean Value Theorem (MVT) Goal: Apply the Mean Value Theorem (MVT) for derivatives to calculate locations where the instantaneous rate of change is equal to the average rate of change. Rolle’s Theorem: Let f be a function that satisfies the following three hypotheses: 1. f is continuous on the closed interval [푎, 푏] 2. f is differentiable on the open interval (푎, 푏) 3. 푓(푎) = 푓(푏) Then there is a number 풄 in (풂, 풃) such that 풇′(풄) = ퟎ Pictures Example 1: Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem. (a) 푓(푥) = 5 – 12푥 + 3푥2, [1,3] (b) 푓(푥) = 푥3– 푥2– 6푥 + 2 , [0,3] The Mean Value Theorem: Let f be a function that satisfies the following two hypotheses: 1. f is continuous on the closed interval [푎, 푏] 2. f is differentiable on the open interval (푎, 푏) 풇(풃)− 풇(풂) Then there is a number 푐 in (푎, 푏) such that 풇′(풄) = or 풇(풃) – 풇(풂) = 풇′(풄)(풃 − 풂) 풃−풂 Why is this important? This means that there is a number at which the instantaneous rate of change (derivative at a point) is equal to the average rate of change over an interval. For example if you are driving your car 150 miles to U of I and it takes 2.5 hours. The average speed you traveled was ________. At some point(s), doesn’t it make perfect sense that your speedometer read _________. Pictures Example 2: Verify that the function satisfies the hypotheses of the M.V.T. on the given interval. Then find all numbers c that satisfy the conclusion of the M.V.T (a) 푓(푥) = 3푥2 + 2푥 + 5, [−1,1] (b) 푓(푥) = 푥3 + 푥 – 1, [0,2] Example 3: Suppose that 푓(0) = −3 and Example 4: Prove that the equation 푥3 + 푥 – 1 = 0 푓′(푥) ≤ 5 for all values of x. How large can 푓(2) has exactly one real root. possible be? AP Calculus 4.3 Local Extreme Values Goal: Apply the first derivative and second derivative tests to determine local extreme values. Increasing/Decreasing Test ′ (a) If 푓 (푥) > 0 on an interval, then f is increasing on that interval (b) If 푓′(푥) < 0 on an interval, then f is decreasing on that interval Example1: Find where the function 푓(푥) = 3푥4 − 4푥3 − 12푥2 + 5 is increasing and where it is decreasing. The First Derivative Test: Suppose that c is a critical number of a continuous function f. (a) If 푓′ changes from positive to negative at c, then f has a local maximum at c. (b) If 푓′ changes from negative to positive at c, then f has a local minimum at c. (c) If 푓′ does not change sign at c, then f has no local maximum or minimum at c. Since 푓′(푥) changes from positive to negative at c, then by the first derivative test, there is a local maximum at c. Since 푓′(푥) changes from negative to positive at c, then by the first derivative test, there is a local minimum at c. Example 2: Find where the local extreme values of 푓(푥) = 3푥4 − 4푥3 − 12푥2 + 5 Example 3: Find where the local extreme values of 푔(푥) = 푥 + 2 sin 푥 0 ≤ 푥 ≤ 2휋 Concavity Test ′′ (a) If 푓 > 0 for all x on an interval, then the graph of f is concave upward on that interval. (b) If 푓′′ < 0 for all x on an interval, then the graph of f is concave downward on that interval. Inflection Point: A point P on a curve 푦 = 푓(푥) is called an inflection point where the graph of f changes concavity. Example 4 : Calculate local extreme values, intervals of concavity & inflection points for the curve 푦 = 푥4 − 4푥3. The Second Derivative Test: Suppose 푓′′ is continuous near c. (a) If 푓′(푐) = 0 and 푓′′(푐) > 0, then f has a local minimum at c. (b) If 푓′(푐) = 0 and 푓′′(푐) < 0, then f has a local maximum at c. Since 푓′(푐) = 0 and 푓′′(푐) < 0 , then by the second derivative test, there is a local maximum at c Since 푓′(푐) = 0 and 푓′′(푐) > 0 , then by the second derivative test, there is a local minimum at c Example 5: Calculate local extreme values of 푓(푥) = 2푥3 + 3푥2 − 36푥 using the second derivative test AP Calculus 4.7 Optimization Goal: Apply the first derivative and second derivative tests to solve optimization problems. First Derivative Test for Absolute Extrema: Suppose c is a critical number of a continuous function f defined on an interval. (a) If 푓′(푥) > 0 for all 푥 < 푐 and 푓′(푥) < 0 for all 푥 > 푐, then 푓(푐) is the absolute maximum value of f. (b) If 푓′(푥) < 0 for all 푥 < 푐 and 푓′(푥) > 0 for all 푥 > 푐, then 푓(푐) is the absolute minimum value of f. Exercise 1: Find two numbers whose sum is 23 and whose Exercise 2: Find two positive numbers whose product is 100 and product is a maximum. whose sum is a minimum. Exercise 3: Find a positive number such that the sum of the Exercise 4: Find two nonnegative numbers whose sum is 9 and so number and its reciprocal are as small as possible that the product of one number and the square of the other number is a maximum. Exercise 5: Find the dimensions of a rectangular pen with area Exercise 6: Build a rectangular pen with three parallel partitions 1000 푚2 whose perimeter is as small as possible using 500 feet of fencing. What dimensions will maximize the total area of the pen? Exercise 7: A farmer has 2400 feet of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? Exercise 8: An open rectangular box with square base is to be made from 48 ft.2 of material. What dimensions will result in a box with the largest possible volume? Exercise 9: Find the point on the parabola 푦2 = 2푥 that is closest to the point (1,4). MAXIMIZING REVENUE A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at $10, the average attendance had been 27,000. When the ticket prices were lowered to $8, the average attendance rose to 33,000. a) Complete the chart. Assume function is linear. Ticket Average Attendance using information Total Revenue Price (try to find pattern) $ $1 $2 $3 b) Find the demand function, assuming that it is linear. $4 $5 $6 c) Find the average attendance using your $7 demand function if tickets were $30 $8 $9 $10 d) Using the demand function, how much would $11 tickets have to cost so that the average attendance was 0? $12 e) Create a revenue function.
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