Solving Concave Functions: Intervals & Inflection Points

[JackieAnne]

Let f(x)=x^6ln(x) . Find (a) the intervals on which f is increasing, (b) the intervals on which f is decreasing, (c) the open intervals on which f is concave up, (d) the open intervals on which f is concave down, and (e) the x-coordinates of all inflection points.

(a) f is increasing on the interval(s)
(b) f is decreasing on the interval(s)
(c) f is concave up on the open interval(s)
(d) f is concave down on the open interval(s)
(e) the x coordinate(s) of the points of inflection are

Notes: In the first four boxes, your answer should either be a single interval, such as [0,1), a comma separated list of intervals, such as (-inf, 2), (3,4], or the word "none".

In the last box, your answer should be a comma separated list of x values or the word "none".


So, I am pretty sure for concave functions we are supposed to find the first and second derivatives.
I am unsure about the first derivative but I got:
(1/7)x(-1/7)*(1/x)
I am unsure on how to get the second derivative from this. Then unsure how to solve the rest of this problem. Thanks!

 

[Dick]

Let f(x)=x^6ln(x) . Find (a) the intervals on which f is increasing, (b) the intervals on which f is decreasing, (c) the open intervals on which f is concave up, (d) the open intervals on which f is concave down, and (e) the x-coordinates of all inflection points.

(a) f is increasing on the interval(s)
(b) f is decreasing on the interval(s)
(c) f is concave up on the open interval(s)
(d) f is concave down on the open interval(s)
(e) the x coordinate(s) of the points of inflection are

Notes: In the first four boxes, your answer should either be a single interval, such as [0,1), a comma separated list of intervals, such as (-inf, 2), (3,4], or the word "none".

In the last box, your answer should be a comma separated list of x values or the word "none".So, I am pretty sure for concave functions we are supposed to find the first and second derivatives.
I am unsure about the first derivative but I got:
(1/7)x(-1/7)*(1/x)
I am unsure on how to get the second derivative from this. Then unsure how to solve the rest of this problem. Thanks!

I'm assuming you mean f(x)=(x^6)*ln(x). Work on your derivative first. It's way wrong. Do you know how to differentiate x^6? Have you heard of the product rule?

 

[JackieAnne]

okay, so I think I got the first derivative:
6x^5*ln(x) + x^6*(1/x)

so then would the second derivative be:

30x^4*ln(x) + x^6*(-1/x^2)

 

[Dick]

okay, so I think I got the first derivative:
6x^5*ln(x) + x^6*(1/x)

so then would the second derivative be:

30x^4*ln(x) + x^6*(-1/x^2)

Ok, you've got the first derivative right, 6*x^5*ln(x)+x^5 if you simplify the second term. Now you are making the same mistake on the second derivative you made on the first. I'll ask you again, have you heard of the product rule? And I'll request that you stop forgetting about it, ok?