The Intermediate and Extreme Value Theorems
Simply stated, if a function is continuous between a low point and a high point, then it must be valued at each intermediate height in between the low and high points.
The converse of an if then statement is a new statement with the hypothesis of the original statement switched with the conclusion of the original statement. In other words, the converse is when the if part of the statement and the then part of the statement are swapped. In general, the converse of a statement is not true.
This statement is false. In order to show the statement is false, all you need is one counterexample where every intermediate value is hit and the function is discontinuous.A counterexample to an if then statement is when the hypothesis (the if part of the sentence) is true, but the conclusion (the then part of the statement) is not true.
First note that the function is a cubic and is therefore continuous everywhere.
Use the Intermediate Value Theorem to show that the following equation has at least one real solution.
This function is continuous because it is the difference of two continuous functions.
Show that there is at least one solution to the following equation.
The function is continuous because it is the sum and difference of continuous functions.
When are you not allowed to use the Intermediate Value Theorem?
The Intermediate Value Theorem should not be applied when the function is not continuous over the interval.
Use the Intermediate Value Theorem to show that each equation has at least one real solution.
15. What do the Intermediate Value and Extreme Value Theorems have to do with continuity?
To see the Review answers, open this PDF file and look for section 14.7.