Continuity and intermediate value theorem

In summary, the continuity theorem states that a function is continuous at a point if the limit of the function at that point is equal to the value of the function at that point, while the intermediate value theorem states that a continuous function must have all values in between two points. These theorems are related in that the continuity theorem is a necessary condition for the intermediate value theorem to hold. They are used in various fields, such as physics and engineering, to model real-world phenomena. Common misconceptions include the belief that a function must be continuous for the intermediate value theorem to apply, and that it can be used to find the exact value of a function at a certain point.
  • #1
dramadeur
19
0
f(x) = x^3 - 12x^2 + 44x - 46

GanLOBR.png

x from 1 to 7

The attempt at a solution:
f(1) = -13
f(2) = 2
f(4) = 2
f(5) = -1
f(6) = 2
So naturally, the answer should be: (1,2) U (4,5) U (5,6)
right? Well, it didn't accept this answer. I think there is something wrong with whatever that is accepting the answer... because it seems to be the correct one.
Even by looking at the graph of the function you'd tell these are the only intervals fitting the criteria...
3KpSros.png

 
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  • #2
Maybe it is the way you have formatted your answer.

AM
 

What is the continuity theorem?

The continuity theorem states that a function is continuous at a point if the limit of the function at that point is equal to the value of the function at that point. In other words, a function is continuous if it has no breaks or jumps in its graph.

What is the intermediate value theorem?

The intermediate value theorem states that if a continuous function has different values at two points, then it must also have all the values in between those two points. This means that a continuous function can't "skip" any values between two points on its graph.

How are the continuity theorem and intermediate value theorem related?

The continuity theorem is a necessary condition for the intermediate value theorem to hold. In other words, a function must be continuous in order for the intermediate value theorem to apply. This is because a function must have no breaks or jumps in its graph in order for it to have all the values in between two points.

How are the continuity and intermediate value theorems used in real life?

These theorems are used in various fields, including physics, engineering, and economics, to model real-world phenomena. For example, in physics, the continuity theorem is used to describe the motion of objects, while the intermediate value theorem is used to analyze the behavior of physical systems.

What are some common misconceptions about the continuity and intermediate value theorems?

One common misconception is that a function must be continuous for the intermediate value theorem to apply. In fact, the intermediate value theorem only requires the function to be continuous between two points, not necessarily at those points. Another misconception is that the intermediate value theorem can be used to find the exact value of a function at a certain point, when in reality it only guarantees the existence of a value between two points.

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