Calculus Name _________________________ 4.1 Maximum and Minimum Reading Assignment Read pages 274 – 278 to help you answer the following. 1. Complete the following statements using the terms absolute maximum, absolute minimum, local maximum, and local

minimum: A function 𝑓 has a(n) ___________________ at 𝑥 = 𝑐 if 𝑓(𝑐) is the smallest function value on the entire domain of 𝑓, whereas 𝑓 has a(n)__________________ at 𝑥 = 𝑐 if 𝑓(𝑐) is the smallest value in some open interval containing c. A function 𝑓 has a(n) ___________________ at 𝑥 = 𝑐 if 𝑓(𝑐) is the greatest function value on the entire domain of 𝑓, whereas 𝑓 has a(n) __________________ at 𝑥 = 𝑐 if 𝑓(𝑐) is the greatest value in some open interval containing c.

The term global maximum or minimum is sometimes used for ________________ or _____________________. Local maximum and local minimum are also called relative maximum and relative minimum. 2. Determine whether each statement is True or False. Draw an example or give an explanation for each that supports

your conclusion. a. The absolute maximum can be the same value as

a local maximum.

b. A function can have multiple absolute minimum values.

c. An absolute maximum or absolute minimum can occur at an endpoint.

d. A local maximum or local minimum can occur at an endpoint.

3. According to the Extreme Value Theorem, what conditions must be true about a function to guarantee that the

function has an absolute maximum and an absolute minimum? 4. Does the Extreme Value Theorem guarantee that the absolute maximum and the absolute minimum exist for the function graphed at the right? Explain.

5. a) What is Fermat’s Theorem? b) Provide an example that shows why the converse of Fermat’s Theorem is not true.

6. Complete the following: A critical number of a function 𝑓 is a number 𝑐 in the domain of 𝑓 such that either ____________________ or __________________.

Rephrase Fermat’s Theorem as: If 𝑓 has a local maximum or local minimum at 𝑐, then _____________________________________________.

7. Read example 7 on Page 278. Why is there a critical number at x = 0? 8. Follow the steps given in the following example. Make sure you understand how to go from Step 2 to Step 3.

Find the critical numbers of the function: 𝑓 𝑥 = !"

!𝑥!! − !"

!𝑥!! + !

!𝑥!!

Step 1: Find 𝑓’(𝑥): 𝑓′ 𝑥 = 8𝑥

!! − 6𝑥

!! + 𝑥

!!

Step 2: Find when 𝑓’(𝑥) = 0 0 = 8𝑥!! − 6𝑥

!! + 𝑥

!!

Step 3: Factor this expression with rational exponents by factoring out the least power of x which is 𝑥

!!.

0 = 𝑥!!(8− 6𝑥 + 𝑥!)

Step 4: Factor the remaining quadratic function and find the zeros. 0 = 𝑥

!!(4− 𝑥)(2− 𝑥)

Zeros: 0, 4 and 2 Step 5: Determine if there are any values where 𝑓’(𝑥) does not exist.

𝑓’(𝑥) exists for all x in its domain.

The critical numbers are 0, 2, and 4. Complete the following from your textbook: Pages 280 - #3–13 odd, 17 – 20, 23, 25, 27, 31, 35, 37* *Refer to the example given above to help with number 37