Intermediate value theorem

In summary: It is a stronger conclusion because it restricts the region in which "c" lies- it gives more information.
  • #1
PFuser1232
479
20
Suppose that ##f## is continuous on ##[a,b]## and let ##M## be any number between ##f(a)## and ##f(b)##.
Then, there exists a number ##c## (at least one) such that:
##a < c < b## and ##f(c) = M##

Why did the author restrict ##c## to ##(a,b)## rather than ##[a,b]##? After all, ##c ∈ [a,b] ⇒ c ∈ (a,b)##.
 
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  • #2
MohammedRady97 said:
After all, ##c ∈ [a,b] ⇒ c ∈ (a,b)##.

False.
 
  • #3
micromass said:
False.
Oops. I just realized it's the other way around.
The question remains; why did the author restrict ##c## to ##(a,b)## rather than ##[a,b]##?
 
  • #4
Because it gives more information.
 
  • #5
MohammedRady97 said:
Suppose that ##f## is continuous on ##[a,b]## and let ##M## be any number between ##f(a)## and ##f(b)##.
Then, there exists a number ##c## (at least one) such that:
##a < c < b## and ##f(c) = M##

Why did the author restrict ##c## to ##(a,b)## rather than ##[a,b]##? After all, ##c ∈ [a,b] ⇒ c ∈ (a,b)##.
You have to be careful here. If "M any number between f(a) and f(b)" includes f(a) and f(b), then you must include the end points a & b in the conclusion ( c is in [a,b] ). Otherwise, if M is properly between f(a) and f(b) but not equal to either, then you can conclude that c is in (a,b). And that is a stronger conclusion that c in [a,b].
 
  • #6
FactChecker said:
You have to be careful here. If "M any number between f(a) and f(b)" includes f(a) and f(b), then you must include the end points a & b in the conclusion ( c is in [a,b] ). Otherwise, if M is properly between f(a) and f(b) but not equal to either, then you can conclude that c is in (a,b). And that is a stronger conclusion that c in [a,b].

Why is it a stronger conclusion? Is it because ##c ∈ [a,b]## implies more obvious conclusions like ##c = a ⇒ f(c) = M = f(a)##?
 
  • #7
It is a stronger conclusion because it restricts the region in which "c" lies- it gives more information. Using "between" to mean "strictly between" is a stronger statement.
 

FAQ: Intermediate value theorem

What is the Intermediate Value Theorem?

The Intermediate Value Theorem is a fundamental theorem in calculus that states that if a continuous function f(x) is defined on a closed interval [a,b], and if y is any number between f(a) and f(b), then there exists at least one number c in the interval [a,b] such that f(c) = y. In simpler terms, the Intermediate Value Theorem guarantees that a continuous function takes on every value between two given points.

Why is the Intermediate Value Theorem important?

The Intermediate Value Theorem is important because it provides a way to prove the existence of a root or solution to an equation, even when it cannot be solved explicitly. It also serves as a key tool in the proofs of other theorems in calculus, including the Mean Value Theorem and the Fundamental Theorem of Calculus.

Can the Intermediate Value Theorem be applied to all functions?

No, the Intermediate Value Theorem only applies to continuous functions. A function is considered continuous if it has no breaks or jumps in its graph, and can be drawn without lifting the pen. If a function is not continuous, the Intermediate Value Theorem cannot be applied.

How is the Intermediate Value Theorem used in real life?

The Intermediate Value Theorem has many real-life applications, especially in fields such as economics, engineering, and physics. It can be used to model and predict the behavior of systems, such as predicting the value of a stock or the temperature of a room, based on known data points. It is also used in computer algorithms to find solutions to equations and optimize functions.

Can the Intermediate Value Theorem be extended beyond one dimension?

Yes, the Intermediate Value Theorem can be extended to higher dimensions. In multivariable calculus, it is known as the Intermediate Value Theorem for Continuous Functions, and states that if a function is continuous on a closed and bounded region in n-dimensional space, and takes on values of both positive and negative, then it must take on the value of 0 at some point within the region.

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