# Showing a Zero of f(x) Using Bolzano's Theorem

• shiloh
It states that if a function is continuous on a closed interval [a, b] and takes on values of opposite signs at the endpoints, then the function must have at least one zero in the interval. In this case, our function f(x)=-3x³+6x²-4x+12 is continuous on the interval [2.2, 2.3] and f(2.2)=-10.336 and f(2.3)=-11.017, which are both negative. Therefore, by the intermediate value theorem, there must exist a value c in the interval [2.2, 2.3] such that f(c)=0. In summary, by using Bolzano's Theorem
shiloh
Use the intermediate value theorem to show that f(x) has a zero in the given interval. Please show all of your work.
f(x)=-3x³+6x²-4x+12;[2.2,2.3]

have to use the Bolzano's Theorem(special case of the intermediate value theorm

shiloh said:
Use the intermediate value theorem to show that f(x) has a zero in the given interval. Please show all of your work.
f(x)=-3x³+6x²-4x+12;[2.2,2.3]

have to use the Bolzano's Theorem(special case of the intermediate value theorm

read the forum rules on how homework questions are to be posted.

Do you know what Bolzano's Theorem states?

## 1. What is Bolzano's Theorem?

Bolzano's Theorem, also known as the Intermediate Value Theorem, states that if a continuous function has different signs at two points, then there must exist at least one point between those two points where the function is equal to zero.

## 2. How do you show a zero of f(x) using Bolzano's Theorem?

In order to show a zero of f(x) using Bolzano's Theorem, you must first determine two points, a and b, where the function has different signs. Then, you can apply the theorem to find a point c between a and b where f(c) = 0.

## 3. Why is Bolzano's Theorem important in mathematics?

Bolzano's Theorem is important in mathematics because it provides a way to prove the existence of solutions to equations and inequalities. It is also used in the proof of the Fundamental Theorem of Calculus, which is a fundamental concept in calculus.

## 4. Can Bolzano's Theorem be applied to all functions?

No, Bolzano's Theorem can only be applied to continuous functions. A continuous function is one that has no sudden jumps or breaks in its graph, and can be drawn without lifting the pencil from the paper. If a function is not continuous, the theorem cannot be applied.

## 5. Are there any limitations to using Bolzano's Theorem?

One limitation of Bolzano's Theorem is that it only guarantees the existence of a zero between two points, but it does not provide an exact value for the zero. Additionally, the theorem assumes that the function is continuous, which may not always be the case in real-world applications. It is important to verify the conditions of the theorem before applying it.

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