Is this answer wrong regarding increacing and decreasing functions?

[The Student]

The question asks, "Let f(x) = x^4 − 32x − 1. Determine on which intervals f is increasing and on which intervals f is decreasing".

The answer has "f is decreasing on (−∞,2] and increasing on (2,∞]".

I have two issues with this answer. One, why would 2 be included on the decreasing side? Two, why is infinity included on one side but not the other; the question did not specify if the function is an element of the reals.

 

[Quaid]

Let f(x) = x^4 − 32x − 1

Determine on which intervals f is increasing and on which intervals f is decreasing

The answer has "f is decreasing on (−∞,2] and increasing on (2,∞]"

why would 2 be included on the decreasing side?

Hi Student:

I think the square bracket in (−∞,2] is a mistake. Both intervals ought to be open intervals. The rate of change at x=2 is zero, so the function is not changing at that point.

Also, it is never correct to include infinity as an endpoint, so the square bracket in (2, ∞] is also a mistake.

Decreasing on (−∞, 2)

Increasing on (2, ∞)

why is infinity included on one side but not the other

I see negative infinity in one and positive infinity in the other. Can you rephrase your question; I'm not sure what you're asking here.

the question did not specify [whether] the function is an element of the reals

It's a polynomial function, so the domain is (−∞, ∞), and the value of f(x) is a Real number for all x.

Cheers :cool:

 

[The Student]

Hi Student:

I think the square bracket in (−∞,2] is a mistake. Both intervals ought to be open intervals. The rate of change at x=2 is zero, so the function is not changing at that point.

Also, it is never correct to include infinity as an endpoint, so the square bracket in (−∞,2] is also a mistake.

Decreasing on (−∞, 2)

Increasing on (2, ∞)




I see negative infinity in one and positive infinity in the other. Can you rephrase your question; I'm not sure what you're asking here.




It's a polynomial function, so the domain is (−∞, ∞), and the value of f(x) is a Real number for all x.

Cheers :cool:

Thank-you very much, your answers are what I was hoping for. :-D

 

[fcabanski]

Thank-you very much, your answers are what I was hoping for. :-D

Where is that answer listed?

 

[The Student]

Where is that answer listed?

Which answer, the poster's answer or my professor's answer?