Determining Intervals 3

[Michael13]

Thanks for the help!!

Determine the intervals on which the following function is increasing and decreasing and classify each of the critical points as a relative minimum, maximum, or neither.

h(x)=3LN(x^3+64)

 

[Quaid]

Determine the intervals on which the following function is increasing and decreasing and classify each of the critical points as a relative minimum, maximum, or neither.

h(x)=3LN(x^3+64)

As in your other threads, you need to post your work so far, on these exercises.

Volunteer tutor need you to provide some idea about why you're confused or uncertain.

What does your textbook have to say about determining intervals in the domain where a function increases?

Have you calculated the derivative of function h, yet?

Please let us know.

Thank you.

 

[Michael13]

Sorry all.

Sorry all, I just wanted to throw them on here real quick. This is what I have so far:

I think I found the h'=(9x^2)/(x^3+64) but I'm not sure if this is right
I know that -4 is not in the domain
I have think I have a critical point of 0 but I am lost from here so any help is really great.
This is a tough problem for me.

 

[srmichael]

Sorry all, I just wanted to throw them on here real quick. This is what I have so far:

I think I found the h'=(9x^2)/(x^3+64) but I'm not sure if this is right
I know that -4 is not in the domain
I have think I have a critical point of 0 but I am lost from here so any help is really great.
This is a tough problem for me.

Your derivative is correct.
Remember, critical points are points where the derivative is either zero OR undefined. You found those as well. h'(x) = 0 at x = 0 and h'(x) is undefined for x= -4 (which, as you mention, is not even in the domain of h(x)). Now just do your sign line. You will find this is a continuously increasing function for x > -4, thus there is no relative minimum or maximum. However, x = 0 is a possible point of inflection.

 

[Michael13]

perhaps I am doing my sign line wrong but I am coming up with increase (-infinity,-4) decrease (-4,0) and increase (0, infinity)

I am using -5, -3, and 3 as my plug in random numbers for 9x^2/x^3+64.

where am I going wrong?

 

[srmichael]

perhaps I am doing my sign line wrong but I am coming up with increase (-infinity,-4) decrease (-4,0) and increase (0, infinity)

I am using -5, -3, and 3 as my plug in random numbers for 9x^2/x^3+64.

where am I going wrong?

First of all, the domain for h(x) is x > -4 so there is no way the function can be increasing for x < -4. For -4 < x < 0, choose test point x = -1.

You get h'(-1) = [9(-1)^2]/[(-1)^3 + 64] = 9/63 > 0 so h'(x) is increasing for -4 < x < 0.

 

[mmm4444bot]

perhaps I am doing my sign line wrong

I am using -5, -3, and 3 as my plug in random numbers for 9x^2/x^3+64

where am I going wrong?

We cannot answer this question because you're not showing your work.