MA 3972-MA-Book April 11, 2018 17:21
138 STEP 4. Review the Knowledge You Need to Score High
8.1 Rolle’s Theorem, Mean Value Theorem, and Extreme
Value Theorem
Main Concepts:Rolle’s Theorem, Mean Value Theorem, Extreme Value Theorem
TIP • Set your calculator to Radians and change it to Degrees if/when you need to. Do not
forget to change it back to Radians after you have finished using it in Degrees.
Rolle’s Theorem
Iff is a function that satisfies the following three conditions:
- fis continuous on a closed interval [a,b]
- fis differentiable on the open interval (a,b)
- f(a)= f(b)= 0
then there exists a numbercin (a,b) such thatf′(c)=0. (See Figure 8.1-1.)
(c, f(c))
f
f′(c) = 0
y
ac b
x
0
Figure 8.1-1
Note that if you change condition 3 fromf(a)=f(b)=0tof(a)= f(b), the conclusion
of Rolle’s Theorem is still valid.
Mean Value Theorem
Iff is a function that satisfies the following conditions:
- fis continuous on a closed interval [a,b]
- fis differentiable on the open interval (a,b)
then there exists a numbercin (a,b) such thatf′(c)=
f(b)−f(a)
b−a
. (See Figure 8.1-2.)