MHB Why Doesn't the Intermediate Value Theorem Apply to ln(x^2 + 2) on [−2, 2]?

[nycmathdad]

Explain why the Intermediate Value Theorem gives no information about the zeros of the function
f(x) = ln(x^2 + 2) on the interval [−2, 2].

Let me see.

Let x = -2.

f(-2) = ln((-2)^2 + 2)

f(-2) = ln(4 + 2)

f(-2) = ln (6). This is a positive value.

When I let x be 2, I get the same answer.

So, f(-2) = f(2).

So, I conclude by saying that the Intermediate Value Theorem does not apply here because both answers are the same value and both are positive.

However, the textbook answer is different.

Textbook Answer:

"This does not contradict the IVT. f (−2) and f (2) are both positive, so there is no guarantee that the function has a zero in the interval (−2, 2)."

 

[jonah1]

Beer soaked ramblings follow.

Explain why the Intermediate Value Theorem gives no information about the zeros of the function
f(x) = ln(x^2 + 2) on the interval [−2, 2].

Let me see.

Let x = -2.

f(-2) = ln((-2)^2 + 2)

f(-2) = ln(4 + 2)

f(-2) = ln (6). This is a positive value.

When I let x be 2, I get the same answer.

So, f(-2) = f(2).

So, I conclude by saying that the Intermediate Value Theorem does not apply here because both answers are the same value and both are positive.

However, the textbook answer is different.

Textbook Answer:

"This does not contradict the IVT. f (−2) and f (2) are both positive, so there is no guarantee that the function has a zero in the interval (−2, 2)."

I see no problem whatsoever.

 

[HOI]

Explain why the Intermediate Value Theorem gives no information about the zeros of the function
f(x) = ln(x^2 + 2) on the interval [−2, 2].

Let me see.

Let x = -2.

f(-2) = ln((-2)^2 + 2)

f(-2) = ln(4 + 2)

f(-2) = ln (6). This is a positive value.

When I let x be 2, I get the same answer.

So, f(-2) = f(2).

So, I conclude by saying that the Intermediate Value Theorem does not apply here because both answers are the same value and both are positive.

However, the textbook answer is different.

Textbook Answer:

"This does not contradict the IVT. f (−2) and f (2) are both positive, so there is no guarantee that the function has a zero in the interval (−2, 2)."

On the contrary the textbook answer is giving the part of YOUR answer that is important! you said "both are positive". The other part of what you said "both answers are the same value" is not important. What if both answers had been 0?

 

[nycmathdad]

On the contrary the textbook answer is giving the part of YOUR answer that is important! you said "both are positive". The other part of what you said "both answers are the same value" is not important. What if both answers had been 0?

I'm curious, what if both answers are 0? I say the Intermediate Value Theorem does not apply.

 

[HOI]

I don't know what you mean by "both answers are 0". The answer to the question is NOT a number, it is an explanation, a sentence. And both you and the textbook say that the explanation is that "both f(2) and f(-2) are positive"- 0 is not between them.

An important point is that this does NOT tell us that there is no zero between -2 and 2. The intermediate value theorem says that IF f(a) is negative and f(b) is positive then there is a zero between a and b. It says nothing at all about what happens if f(a) and f(b) are both positive or both negative. There might be a zero between a and b or there might not.

 

[nycmathdad]

I don't know what you mean by "both answers are 0". The answer to the question is NOT a number, it is an explanation, a sentence. And both you and the textbook say that the explanation is that "both f(2) and f(-2) are positive"- 0 is not between them.

An important point is that this does NOT tell us that there is no zero between -2 and 2. The intermediate value theorem says that IF f(a) is negative and f(b) is positive then there is a zero between a and b. It says nothing at all about what happens if f(a) and f(b) are both positive or both negative. There might be a zero between a and b or there might not.

Here is what you said:

"On the contrary the textbook answer is giving the part of YOUR answer that is important! you said "both are positive". The other part of what you said "both answers are the same value" is not important. What if both answers had been 0?"

Read your question at the end.