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Math 2413
Section 1.5 – The Intermediate Value Theorem
Section 1.5 Notes
Theorem 1.5.1: The Intermediate Value Theorem
If f is a continuous function on the closed interval [a,b], and N is a real number such that
f (a) ≤ N ≤ f (b) or f (b) ≤ N ≤ f (a), then there is at least one number c in the interval (a,b) such that f (c) = N .
That is, if a function is continuous on a closed interval, it does not skip any values; it has no breaks or jumps. In
geometric terms, this theorem says that if y = N is a horizontal line between y = f (a) and y = f (b), then the
graph of f must intersect y = N somewhere (at least once).
There is a hole on the graph of function f(x) at x = -2. We say this function is NOT continuous at x = -2.
This theorem can be used to prove the existence of the zeros of a function. If f (a) < 0 and f (b) > 0 (or vice
versa), then
f (a) < 0 < f (b) and by the Intermediate Value Theorem, there is at least one c such that f (c) = 0 .
That is, the function has at least one root between the numbers a and b.
f (a) < 0 < f (b)
f (b) < 0 < f (a)
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Math 2413
Example 1: Show that the equation x 3  x 2  5 x  2  0 has a root between 0 and 1.
Section 1.5 Notes
Theorem 1.5.2: The Extreme-Value Theorem
If f is continuous on a bounded interval [a,b], then f takes on both a maximum value and a minimum value.
Examples: For each function, determine whether the function attains the extreme value on the specified
interval.
1.
f ( x )  x on its domain.
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Math 2413
2.
Section 1.5 Notes
if x  0
 5

f ( x)   x if 0  x  3 on the interval [0, 3]
5
if 3  x

Examples:
 x2
if
1. Find A given that the function f ( x)  
 Ax  42 if
x6
x6
is continuous at 6.
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Math 2413
Section 1.5 Notes
2. In Exercises 15-26, state whether it is possible to have a function f defined on the indicated:
a. f is defined on [2, 5]; f is continuous on [2, 5], minimum value 2, maximum value f (2) = 5
b. f is defined on [4, 5]; f is continuous on [4, 5) , minimum value f (5) = 4, and no maximum value.
c. f is defined on [3, 6]; f is continuous on [3, 6], maximum value of 6 and a minimum value of 6.
d. f is defined on [2, 5]; f is continuous on [2, 5], is non-constant, and takes on only integer values.
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