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- 3.3: Intermediate Value Theorem, Existence of Solution
- MTH 102 Analysis in One Variable
- Worked example: using the intermediate value theorem
- Intermediate Value Theorem

## 3.3: Intermediate Value Theorem, Existence of Solution

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In this section we want to take a look at the Mean Value Theorem. In most traditional textbooks this section comes before the sections containing the First and Second Derivative Tests because many of the proofs in those sections need the Mean Value Theorem. First, we should show that it does have at least one real root. We now need to show that this is in fact the only real root. Since this assumption leads to a contradiction the assumption must be false and so we can only have a single real root. Here is the theorem. We can see this in the following sketch.

Can the same be said for the function:? Intasar Maths Teacher 4. Meet all our tutors Dr. Kritaphat Maths Teacher 4. Meet all our tutors Matthew Maths Teacher 4. Meet all our tutors Petar Maths Teacher 4. Meet all our tutors Tamsyn-clara Maths Teacher 5.

## MTH 102 Analysis in One Variable

Continuous is a special term with an exact definition in calculus, but here we will use this simplified definition:. Imagine we are rotating the table , and the 4th leg could somehow go into the ground like sand :. So there must be some point where the 4th leg perfectly touches the ground and the table won't wobble. The famous Martin Gardner wrote about this in Scientific American. There is also a very complicated proof somewhere. At some point during a round-trip you will be exactly as high as where you started. Hide Ads About Ads.

Example (from the textbook). Use the Intermediate Value Theorem to show that there is root of the equation. 4x3 − 6x2 + 3x − 2=0 in the interval [1, 2]. Solution: Consider the function f(x)=4x3−6x2+3x−2 over the closed interval [1, 2].

## Worked example: using the intermediate value theorem

Many functions have the property that their graphs can be traced with a pencil without lifting the pencil from the page. Normally, such functions are called continuous. Other functions have points at which a break in the graph occurs, but satisfy this property over intervals contained in their domains.

*It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper. Remark: Version II states that the set of function values has no gap. A subset of the real numbers with no internal gap is an interval.*

### Intermediate Value Theorem

The Intermediate Value Theorem is one of the most important theorems in Introductory Calculus, and it forms the basis for proofs of many results in subsequent and advanced Mathematics courses. Generally speaking, the Intermediate Value Theorem applies to continuous functions and is used to prove that equations, both algebraic and transcendental , are solvable. The formal statement of this theorem together with an illustration of the theorem follow. All functions are assumed to be real-valued. Click HERE to see a detailed solution to problem 1. Click HERE to see a detailed solution to problem 2. Click HERE to see a detailed solution to problem 3.

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#### More Formal

Халохот услышал, как где-то ниже тело Беккера упало на каменные ступеньки, и бросился вниз, сжимая в руке пистолет. В поле его зрения попало окно. Здесь. Халохот приблизился к внешней стене и стал целиться. Ноги Беккера скрылись из виду за поворотом, и Халохот выстрелил, но тут же понял, что выстрел пришелся в пустоту. Пуля срикошетила от стены.

Хейл понимал: то, что он сейчас скажет, либо принесет ему свободу, либо станет его смертным приговором. Он набрал в легкие воздуха. - Вы хотите приделать к Цифровой крепости черный ход. Его слова встретило гробовое молчание. Хейл понял, что попал в яблочко.

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The conclusions may be obvious when you understand the statements and look at a graph, but they are powerful nonetheless.

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