How Does the Intermediate Value Theorem Confirm a Unique Zero in an Interval?

In summary, the function f(x) = x^4 + 3x + 1 has exactly one zero in the interval [-2, -1]. This means that there is only one value of x that makes the function equal to zero within that interval. To find this zero, you can use the intermediate value theorem by evaluating the function at the endpoints of the interval and determining if the function changes sign between them. If it does, then there must be a zero somewhere between the two points.
  • #1
donjt81
71
0
Can someone get me started on this problem.

Show that the function has exactly one zero in the given interval.
f(x) = x^4 + 3x + 1 [-2, -1]

I know I have to use the Intermediate Value theorem in this but not really sure how to apply it. Also what does it mean when it says "a function has one zero" what is a zero?

Thanks in advance
 
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  • #2
A zero is a number x such that f(x)=0. For example, the function [tex]f(x)=x^2 - 1 = (x-1)(x+1)[/tex] has the zeros 1 and -1.

For the function above, it is hard to tell by some formula whether it has a real zero (not all functions have real zeros, and many have no zeros). But it is easy to find points where the function is negative and points where the function is positive. What does the intermediate value theorem tell you then?
 
  • #3
donjt81 said:
Can someone get me started on this problem.

Show that the function has exactly one zero in the given interval.
f(x) = x^4 + 3x + 1 [-2, -1]

I know I have to use the Intermediate Value theorem in this but not really sure how to apply it. Also what does it mean when it says "a function has one zero" what is a zero?

Thanks in advance
What is f(-2)? What is f(-1)? What does that tell you (using the intermediate value theorem)?
 

FAQ: How Does the Intermediate Value Theorem Confirm a Unique Zero in an Interval?

What is the Intermediate Value Theorem?

The Intermediate Value Theorem is a mathematical theorem that states that if a continuous function takes on two values at two points of its domain, it must also take on every value between those two points.

What is the importance of the Intermediate Value Theorem?

The Intermediate Value Theorem is important because it allows us to prove the existence of roots or solutions to equations, even when we cannot find them explicitly. It also helps us to understand the behavior of continuous functions and their graphs.

How is the Intermediate Value Theorem applied in real life?

The Intermediate Value Theorem has many real-life applications, such as in economics, physics, and engineering. For example, it can be used to prove the existence of solutions to optimization problems or to model the behavior of a system over time.

Can the Intermediate Value Theorem be extended to multivariable functions?

Yes, the Intermediate Value Theorem can be extended to multivariable functions, but it is known as the Intermediate Value Theorem for Paths. It states that if a continuous function takes on two values at two points connected by a continuous path, then it must also take on every value between those two points along that path.

Are there any conditions for the Intermediate Value Theorem to hold?

Yes, there are two conditions for the Intermediate Value Theorem to hold. The first condition is that the function must be continuous on the closed interval between the two points. The second condition is that the function must take on two different values at the two points. If both conditions are met, then the Intermediate Value Theorem holds.

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