University:
Glendale Community College
Course:
MAT142 | College Mathematics
Academic year:

2020
Views:

168

Pages:

18
Author:

Maddison Riddle

Section 4.1 Absolute Extrema (Minimum Homework: 1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25) We cover the concept of the absolute maximum and absolute minimum in section 4.1. Given a graph of a continuous function over a closed interval: (We only consider graphs that are bound on each side when we find absolute maximum and absolute minimum values.) • Absolute maximum: y-coordinate of the highest point • Absolute minimum: y-coordinate of the lowest point. The absolute maximum and absolute minimum points are marked on the graph below. It would be proper to write the following: • There is an absolute maximum of y = 2, which occurs when x = 2. • There is an absolute minimum of y = -2, which occurs when x = 0. (It would be correct to write 𝑓(𝑥 ) 𝑖𝑛𝑠𝑡𝑒𝑎𝑑 𝑜𝑓 𝑦) . We need to be able to find the absolute maximum and absolute minimum of values of a function over a given interval [a,b]. Here are the steps: 1) Find 𝑓′(𝑥) 2) Solve 𝑓 ′ (𝑥 ) = 0 3) Create a table with two columns. 𝑥 Left column a b Any answer from step 2 in the interval [a,b] There will be as many extra rows as answers to part 2 in the given interval [a,b] 𝑓(𝑥 ) Right column 4) Substitute the values in the left column into the ORIGINAL function to complete the right column. 𝑥 Left column a b Any answer from step 2 in the interval [a,b] 𝑓(𝑥 ) Right column 𝑓(𝑎) 𝑓(𝑏) 𝑓(#) 5) Write Answer Absolute maximum – the largest value in the right column Absolute minimum – smallest value in the right column. Example: 𝑓(𝑥 ) = 𝑥 3 − 3𝑥 2 ; [1,4] Find the absolute maximum and absolute minimum of the function under the given interval. 1) Find 𝑓′(𝑥) 𝑓 ′ (𝑥 ) = 3𝑥 2 − 6𝑥 2) Solve 𝑓 ′ (𝑥 ) = 0 3𝑥 2 − 6𝑥 = 0 3𝑥 (𝑥 − 2) = 0 3𝑥 = 0 𝑥−2=0 𝑥=0 𝑥=2 3) Create a table with two columns. Only include 1,2 𝑎𝑛𝑑 4 𝑖𝑛 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑐𝑜𝑙𝑢𝑚𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑏𝑙𝑒. 𝐷𝑜 𝑛𝑜𝑡 𝑖𝑛𝑐𝑙𝑢𝑑𝑒 0 𝑎𝑠 𝑖𝑡 𝑖𝑠 𝑛𝑜𝑡 𝑏𝑒𝑡𝑤𝑒 𝑥 1 2 4 𝑓(𝑥) 𝑥 1 2 4 𝑓(𝑥) 𝑓(1) = (1 − 3(1)2 = −2 𝑓(2) = (2)3 − 3(2)2 = −4 𝑓 (4) = (4)3 − 3(4)2 = 16 4) )3 5) Write Answer absolute maximum of y = 16, which occurs when x = 4. absolute minimum of y = -4, which occurs when x = 2. (Minimum Homework: 1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25) #1-9: Find the absolute maximum and absolute minimum 1) 2) Absolute maximum of 𝑦 = 5, which occurs when 𝑥 = 1. Absolute minimum of 𝑦 = 1, which occurs when 𝑥 = 3. (Minimum Homework: 1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25) 3) 4) Absolute maximum of 𝑦 = 5, which occurs when 𝑥 = 2. Absolute minimum of 𝑦 = −4, which occurs when 𝑥 = 5. (Minimum Homework: 1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25) 5) 6) Absolute maximum of 𝑦 = 10, which occurs when 𝑥 = −2, 2. Absolute minimum of 𝑦 = −6, which occurs when 𝑥 = 0. (Minimum Homework: 1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25) 7) 8) Absolute maximum of 𝑦 = 8, which occurs when 𝑥 = 2. Absolute minimum of 𝑦 = −1, which occurs when 𝑥 = −1. (Minimum Homework: 1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25) 9) (Minimum Homework: 1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25) #10-27: Find the absolute maximum and absolute minimum of the function under the given interval. 11) 𝑓(𝑥 ) = 𝑥 2 − 6𝑥 + 4; [−5,5] 10) 𝑓(𝑥 ) = 𝑥 2 − 2𝑥 + 5; [−3,3] 1) Find 𝑓′(𝑥) 2) Solve 𝑓 ′ (𝑥 ) = 0 3) Create a table with two columns. 𝑥 𝑓(𝑥) 𝑥 𝑓(𝑥) 4) 5) Write Answer Absolute maximum of 𝑦 = 20, which occurs when 𝑥 = −3. Absolute minimum of 𝑦 = 4, which occurs when 𝑥 = 1. 13) 𝑓(𝑥 ) = 𝑥 3 + 6𝑥 2 ; [−2,1] 12) 𝑓(𝑥 ) = 𝑥 3 − 6𝑥 2 ; [−2, 2] 1) Find 𝑓′(𝑥) 2) Solve 𝑓 ′ (𝑥 ) = 0 3) Create a table with two columns. 𝑥 𝑓(𝑥) 𝑥 𝑓(𝑥) 4) 5) Write Answer Absolute maximum of 𝑦 = 0, which occurs when 𝑥 = 0. Absolute minimum of 𝑦 = −32, which occurs when 𝑥 = −2. 14) 𝑓(𝑥 ) = 𝑥 3 − 3𝑥 2 ; 3𝑥 2 + 2; [−1,5] 15) 𝑓(𝑥 ) = 𝑥 3 − [−1,3] 17) 𝑓(𝑥 ) = 3𝑥 4 − 4𝑥 3 ; [-2,3] 16) 𝑓(𝑥 ) = 𝑥 4 − 𝑥 3 + 5; [-2,2] 1) Find 𝑓′(𝑥) 2) Solve 𝑓 ′ (𝑥 ) = 0 3) Create a table with two columns. 𝑥 𝑓(𝑥) 𝑥 𝑓(𝑥) 4) 5) Write Answer Absolute maximum of 𝑦 = 29, which occurs when 𝑥 = −2. Absolute minimum of 𝑦 = 1253 256 3 , which occurs when 𝑥 = . 4 19) 𝑓(𝑥 ) = (𝑥 2 − 16)3 ; [-2,2] 18) 𝑓(𝑥 ) = (𝑥 2 − 9)4 ; [0,2] 1) Find 𝑓′(𝑥) 2) Solve 𝑓 ′ (𝑥 ) = 0 3) Create a table with two columns. 𝑥 𝑓(𝑥) 𝑥 𝑓(𝑥) 4) 5) Write Answer Absolute maximum of 𝑦 = 6561, which occurs when 𝑥 = 0. Absolute minimum of 𝑦 = 625, which occurs when 𝑥 = 2. 3 20) 𝑓(𝑥 ) = √𝑥 ; [−1,2] 5 21) 𝑓(𝑥 ) = √𝑥 ; [−3,2] 23) 𝑓(𝑥 ) = 2𝑥𝑒 𝑥 ; [0,3] 22) 𝑓(𝑥 ) = 𝑥𝑒 𝑥 ; [−3,3] 1) Find 𝑓′(𝑥) 2) Solve 𝑓 ′ (𝑥 ) = 0 3) Create a table with two columns. 𝑥 𝑓(𝑥) 𝑥 𝑓(𝑥) 4) 5) Write Answer Absolute maximum of 𝑦 = 3𝑒 3 , which occurs when 𝑥 = 3. Absolute minimum of 𝑦 = −1/𝑒, which occurs when 𝑥 = −1. 2 25) 𝑓(𝑥 ) = 𝑒 𝑥 ; [−2,1] 2 24) 𝑓(𝑥 ) = 𝑒 3𝑥 ; [−1,1] 1) Find 𝑓′(𝑥) 2) Solve 𝑓 ′ (𝑥 ) = 0 3) Create a table with two columns. 𝑥 𝑓(𝑥) 𝑥 𝑓(𝑥) 4) 5) Write Answer Absolute maximum of 𝑦 = 𝑒 3 , which occurs when 𝑥 = −1, 1. Absolute minimum of 𝑦 = 1, which occurs when 𝑥 = 0. 26) 𝑓(𝑥 ) = 𝑥 3 𝑒 𝑥 ; [-3,1] 27) 𝑓(𝑥 ) = 𝑥 2 𝑒 𝑥 ; [-3,1]