Intermediate Value Theorem

Intermediate Value Theorem is a theorem in calculus which defines an important property of continuous functions. It is abbreviated as IVT. The theorem is quite intuitive one but provides a significant result for the interpretation of the behaviour of functions. It can be used to know the range of values for a physical quantity such as temperature if an expression in terms of time or other variables is known for it. Other applications of the theorem include solving equations, proving the existence of roots, and analyzing real-world problems where continuity is observed.

In this article, we will learn the statement of the theorem, its proof by two different approaches, its various applications, the converse of the theorem, some numerical problems and related frequently asked questions.

Intermediate Value Theorem also called IVT is a theorem in calculus about values that continuous functions attain between a defined interval. It provides a formal statement for the intuitive understanding of continuous functions. For instance, if we drive from one city to another where the starting point is at sea level and the ending point is at some higher elevation. The theorem supports the fact that at some point in the journey, we must pass through every intermediate elevation. The formal statement of the theorem is stated as follows.

Statement of the Intermediate Value Theorem

The Intermediate Value Theorem states that,

For a continuous function f defined on the closed interval [a,b], where f(a) ≠ f(b) and d is real number between f(a) and f(b), then there exists at least one real number c between in the interval (a,b) such that f(c) = d.

The theorem is also stated for a specific case when f(a) and f(b) have opposite signs, then there exists at least one real number c between a and b such that f(c) = 0. This statement helps to get inference about roots of the function.

The theorem can be proved by two common approaches. Both are discussed one by one as follows.

Proof of IVT Based on Real Numbers’ Completeness

The completeness of real numbers is the property which makes real numbers distinguishable from rational numbers and natural numbers. It implies that in a given interval of real numbers, each number is a real number. This property is understood in different aspects, the concept of a supremum (least upper bound) being the most common one is used to prove the theorem. The supremum property states that for any set S which is subset of R which has an upper bound, there exists a smallest real number sup S that is greater than or equal to every element in S.

Now, we have a continuous function f defined on the interval [a, b], and d lies between f(a) and f(b). To prove that that there exists c ∈ [a, b] such that f(c) = d, we define a set as follows,

S = {x ∈ [a, b] | f(x) ≤ d}

Now, the upper bound of set S is b, as x ≤ b for all x∈[a, b]. By the completeness property of the real numbers, S has a supremum, say c = sup S, which implies that c ∈ [a,b]. We need to prove that f(c) = d.

Let f(c) ≠ d, then two cases arise,

• If f(c) < d, it implies that there exists an ?>0 such that for all x∈(c, c+?)∩[a, b], f(x)<d. This contradicts c = sup S as it implies there are points greater than c but still in S.
• If f(c) > d, it implies that there exists an ?>0 such that for all x∈(c-?, c)∩[a, b], f(x)<d. This contradicts c = sup S as it implies f takes values less than d for points close to c from the left.

From the contradictions proved above, it comes out that f(c) = d. Hence, we have proved that there exists c ∈ [a, b] such that f(c) = d, which proves the Intermediate Value Theorem.

Proof of IVT Using Continuity and Supremum

The theorem can also be proved using the property of continuity and supremum. The proof is similar to the previous one focusing on properties of continuous functions and supremum of a set.

Let us define a set S as follows,

S = {x ∈ [a, b] | f(x) ≤ d}

Let the supremum of set S is given as follows,

c = sup S

where,

c ∈ [a, b]

Now, let f(c) ≠ d, then two cases arise,

• f(c) < d, it implies that there exists a ?>0 such that if ∣x – c∣<?, then ∣f(x)−f(c)∣<d−f(c). But this would imply that f(x)<d for x ∈ (c ,c+?) ∩ [a, b], which contradicts c = sup S as there would be points greater than c with f(x) ≤ d.
• f(c) > d, it implies that there exists a ?>0 such that if ∣x – c∣<?, then ∣f(x)−f(c)∣<f(c)-d, but this would imply that f(x)>d for x ∈ (c−?, c) ∩ [a, b]. Here, x < c and c = sup S, which means f(x) ≤ d that contradicts the possibility of f(x) > d.

We get f(c) = d from the contradictions proved above, which proves the Intermediate Value Theorem (IVT).

Various applications of the Intermediate Value Theorem are discussed as follows:

• The theorem can be used to check the existence of roots of a continuous function in a specified interval.
• The theorem helps to check whether a continuous function attains a given value in a defined interval.
• The theorem is used in solving equations through numerical methods such as Bisection Method.
• The theorem serves as a base for proving several other theorems in calculus such as Mean Value Theorem.
• The theorem is also used to find critical points, i.e. points where the derivative of the function attains a specific value.

Although, Intermediate Value Theorem has various applications but it has some limitations too. These are discussed as follows:

• The theorem does not provide any information on number of roots in the interval for the function.
• The theorem is applicable to continuous functions only and is uncertain for discontinuous or piecewise functions.
• The theorem does not give any information about the behaviour of function outside the specified interval.

The converse of the Intermediate Value Theorem (IVT) is not always true. The converse statement is stated as follows:

If there exists a point c ∈ [a, b] such that f(c)=d for every number d between f(a) and f(b), then f is continuous for the interval [a, b].

The above statement is not true always. A function can follow the Intermediate Value Theorem despite being discontinuous. In other words, a function following the IVT property need not to be a continuous function but a continuous function always follows the Intermediate Value Theorem.

Also, Check

• Mean Value Theorem
• Cauchy’s Mean Value Theorem
• Continuity of Functions

Example 1: Check whether the function defined as f(x) = x³ – 8 has a root in the interval [0,4].

Solution:

Here, we have,

f(0) = 0 – 8 = -8, and

f(4) = 4³ – 8 = 64 – 8 = 56.

As the function yields values with opposite signs at the endpoints of the given interval, by intermediate value theorem, it implies that the function has at least one root in the interval.

Example 2: Show that the function defined as f(x) = e^x – 3x has a root in the interval [0,1].

Solution:

To show whether the function has a root in the given interval, we check the value of function at the endpoints of the interval,

We have, f(0) = e⁰ – 3(0) = 1 – 0 = 1, and

f(1) = e – 3 = -0.28 (approx.)

Thus, we see that function has opposite signed values at endpoints of the interval, so it has at least one root in the interval.

Q1: Check whether the function defined as f(x) = x² – 2x has a root in the interval [0, 1].

Q2: Show that the function defined by f(x) = 1 – 2sin(x) has at least one root in the interval [0, π/2].

Q3: Consider the function f(x) = x³ – x + 2, check whether it has a root in the interval [1, 4].

Q4: Check whether the function given by f(x) = 4x – e^x has a root in the interval [0, 1].

Q5: Show that the function defined as f(x) = x⁵ – x has at least one root in the interval [-1, 1].

What is meant by the Intermediate Value Theorem?

The Intermediate Value Theorem states that a continuous function attains every intermediate value in an interval specified by functional values of the function on two input variables in the domain of the function.

What is the necessary condition for the Intermediate Value Theorem to be applicable?

The function should be continuous in the given interval for Intermediate Value Theorem to be applicable.

How is the Intermediate Value Theorem applicable in the field of physics?

Intermediate Value Theorem can be used to check whether a particular value would be attained by a physical quantity expressed in form of a function in a specified interval.

Discuss some cases where Intermediate Value Theorem can not be applied.

Intermediate Value Theorem is applicable to continuous functions but it can not be applied to discontinuous or piecewise functions.

Does the Intermediate Value Theorem apply if functional values at endpoints of the interval are equal?

Yes, the theorem is applicable if functional values at endpoints are equal too.

Does Intermediate Value Theorem give any information about number of roots in the specified interval?

No, the theorem checks the presence or absence of roots in the specified interval but does not gives any information about number of roots.