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Flashcards: Apply Intermediate Value Theorem

Which of the following does NOT satisfy the conditions required to apply the Intermediate Value Theorem to a function $\displaystyle f$ on the interval [a,b] $\displaystyle [a,b]$ ?

The function f must be continuous along [a,b] $\displaystyle [a,b]$ to apply Intermediate Value Theorem

Intermediate Value Theorem can be used when $\displaystyle f(a) = f(b)$

Intermediate Value Theorem considers points within the interval [a,b] $\displaystyle [a,b]$

Intermediate Value Theorem cannot be applied outside of the interval [a,b] $\displaystyle [a,b]$

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Flashcards: Apply Intermediate Value Theorem

Which of the following does NOT satisfy the conditions required to apply the Intermediate Value Theorem to a function $\displaystyle f$ on the interval [a,b] $\displaystyle [a,b]$ ?

The function f must be continuous along [a,b] $\displaystyle [a,b]$ to apply Intermediate Value Theorem

Intermediate Value Theorem can be used when $\displaystyle f(a) = f(b)$

Intermediate Value Theorem considers points within the interval [a,b] $\displaystyle [a,b]$

Intermediate Value Theorem cannot be applied outside of the interval [a,b] $\displaystyle [a,b]$

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Flashcards: Apply Intermediate Value Theorem

Which of the following does NOT satisfy the conditions required to apply the Intermediate Value Theorem to a function $\displaystyle f$ on the interval [a,b] $\displaystyle [a,b]$ ?

Your answer:

The function f must be continuous along [a,b] $\displaystyle [a,b]$ to apply Intermediate Value Theorem

Your answer:

Intermediate Value Theorem considers points within the interval [a,b] $\displaystyle [a,b]$

Your answer:

Intermediate Value Theorem cannot be applied outside of the interval [a,b] $\displaystyle [a,b]$

Correct answer:

Intermediate Value Theorem can be used when $\displaystyle f(a) = f(b)$

Explanation:

Intermediate Value Theorem states that if the function $\displaystyle f$ is continuous and has a domain containing the interval [a,b] $\displaystyle [a,b]$ , then at some number $\displaystyle c$ within the interval [a,b] $\displaystyle [a,b]$ the function will take on a value f(c)=N $\displaystyle f(c)=N$ that is between the values of f(a) $\displaystyle f(a)$ and f(b) $\displaystyle f(b)$ .

The conditions that must be satisfied in order to use Intermediate Value Theorem include that the function must be continuous and the number $\displaystyle c$ must be within the interval [a,b] $\displaystyle [a,b]$ . However, f(a) $\displaystyle f(a)$ cannot equal f(b) $\displaystyle f(b)$ .

Therefore, the answer choice “Intermediate Value Theorem can be used when $\displaystyle f(a) = f(b)$ ” does not satisfy the necessary conditions and is the correct answer for this question.

Flashcards: Apply Intermediate Value Theorem

Using Intermediate Value Theorem to analyze a continuous function, what can be deduced if a polynomial changes signs within an interval?

There is a zero between the endpoints of that interval

The function is not differentiable within that interval

The function is not continuous within that interval

There is a local maximum between the endpoints of that interval

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Flashcards: Apply Intermediate Value Theorem

Using Intermediate Value Theorem to analyze a continuous function, what can be deduced if a polynomial changes signs within an interval?

Your answer:

The function is not differentiable within that interval

Your answer:

The function is not continuous within that interval

Your answer:

There is a local maximum between the endpoints of that interval

Correct answer:

There is a zero between the endpoints of that interval

Explanation:

Intermediate Value Theorem is only true with continuous, differentiable functions, thus eliminating the answer choices “The function is not continuous within that interval” and “The function is not differentiable within that interval.” There is not necessarily a local maximum or minimum contained in the interval either. That leaves the correct answer choice, “There is a zero between the endpoints of that interval.” If the polynomial is changing signs and meets the requirements for Intermediate Value Theorem, it must cross the $\displaystyle x$ axis at some point within the interval.

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