1.4 Intermediate Value Theorem
Intermediate Value Theorem:
If f is continuous on the closed interval [a, b] and k is any number between f(a) and f(b),
then there is at least one number c in [a, b] such that f(c) = k.
An informal application of the Intermediate Value Theorem would be the following: If
you were 2 ! feet tall when you were 3 years old and you were 6 feet tall when you were
16 years old, then there must have been some time when you were 5 feet tall. What is
[a, b], what is c and what is k for this application?
Homework Example:
1. Verify that the Intermediate Value Theorem applies to the indicated interval and find
the value of c guaranteed by the theorem.
f (x) = x 2 ! 6x + 8
[0, 3] f (c) = 0
Solution: Make sure that the conditions of the Intermediate Value Theorem are
met.
• Continuous on [0, 3] i.e., continuous on [a, b]
• 0 lies between f(0) and f(3) i.e., f(a) < f(c) < f(b) or f(a) > f(c) > f(b)
f(0) = 02 – 6(0) + 8 = 8
f(3) = 32 – 6(3) + 8 = -1
-1 < 0 < 8
• Intermediate Value Theorem applies
0 = x 2 ! 6x + 8
0 = (x ! 4)(x ! 2)
x = 4 and x = 2
c = 2 (4 is outside the interval [0,3] )
f(x) = (x2-6!x)+8
6
4
2
-10
-5
5
-2