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

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SUMMARY

The discussion focuses on applying the Intermediate Value Theorem (IVT) to demonstrate that the function f(x) = x^4 + 3x + 1 has exactly one zero in the interval [-2, -1]. Participants clarify that a zero of a function is a value x for which f(x) = 0. By evaluating f(-2) and f(-1), users can establish the function's behavior at these endpoints, confirming the existence of a zero within the specified interval due to the change in sign of the function values.

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  • Understanding of the Intermediate Value Theorem (IVT)
  • Basic knowledge of polynomial functions
  • Ability to evaluate functions at specific points
  • Familiarity with the concept of function zeros
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  • Learn how to apply the Intermediate Value Theorem in various contexts
  • Explore polynomial function properties and their zeros
  • Study graphical methods for identifying function behavior
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Students studying calculus, educators teaching mathematical analysis, and anyone interested in understanding the application of the Intermediate Value Theorem in confirming function zeros.

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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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A zero is a number x such that f(x)=0. For example, the function f(x)=x^2 - 1 = (x-1)(x+1) 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?
 
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)?
 

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