104 STEP 4. Review the Knowledge You Need to Score High
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 7.1-1.)
(c, f (c))
f
f′ (c)=0
y
ac b
x
0
Figure 7.1-1
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 7.1-2.)
(c,f(c))
(a,f(a))
(b,f(b))
c x
y
f
0
f′(c) = f(b) – f(a)
b – a
Figure 7.1-2