What is Intermediate Value Theorem

Let's say you checked the continuity of the function of closed intervals and you found that the function is continuous. That is not all, you also need to check whether the function passes through the expected points. We call that expected point an Intermediate Value. For example, a function proves to be continuous on ending points, "a" and "b". A point, "c", is somewhere between both points. If the graph is continuous, it should pass through point "c" as well. To prove that, we use a theorem which is called the Intermediate Value Theorem. This theorem says, "If a function is continuous on the closed interval [a, b] and k is any number between f(a) and f(b) then there exists a number, c, within (a, b) such that f(c) = k."

By observing the graph, the intermediate value theorem can be defined another way:

  • If a function is continuous on the closed interval [a, b], the function takes all values between f(a) and f(b) in this interval.
  • The intermediate value theorem does not indicate the value or values of c, it only determines their existence.

You must be wondering why the intermediate value theorem is important? Because it provides valuable data. This theorem can help in the evaluation of the function at a specific point. Furthermore, it can be used to prove the existence of the roots of an equation.

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Example

Prove that the image of 2 exists in function f(x) = x (\sin { x } + 1).

The function is continuous in \mathbb{R} as it is the product of two continuous functions.

Take the interval [- \frac { \pi }{ 2 }, \frac { \pi }{ 2 } ], and study the value of the extremes:

f(- \frac { \pi }{ 2 } ) = 0 < 2

f(\frac { \pi }{ 2 } ) = \pi > 2

Therefore there is a c \in (- \frac { \pi }{ 2 }, \frac { \pi }{ 2 }) such that f(c) = 2.

 

Prove that the image of 7 exists in function f(x) = { x }^{ 2 } with intervals [2, 3].

The function is continuous in \mathbb{R} as it is the product of two continuous functions.

Studying extreme values:

f(2) = 4 < 7

f(3) = 9 > 7

Therefore there is a c \in [2, 3] such that f(c) = 7.

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.