Question

f(x) = (2x +1)^3

f'(x) = 6(2x + 1)^2

f''(x) = 48x + 24

I need to know when its concave up/down increasing /decreasing and the inflection points I am new to this kind of stuff

Answer 1

First derivative is nonnegative for all real x, so f is non-decreasing.

Second derivative is everywhere matching the sign of x+1/2, so there is an inflection point at x=-1/2. The function is concave down on x<-1/2 and concave up on x>-1/2.

Answer 2

how did you find the -1/2

Answer 3

f''(x)=0 at x=-1/2

Answer 4

Ok and something more
are my derivative calculations correct?

f(x) = (2x +1)^3

f'(x) = 6(2x + 1)^2

Answer 5

Yes, all were perfect!

Answer 6

so its not decreasing that means its always increasing? Kinda what's the interval?

Answer 7

(0,infinity) increasing?

Answer 8

non-decreasing means increasing or flat. it is flat at the inflection point, increasing everywhere else

Answer 9

so increasing on the entire real line except at -1/2, where it is flat (deriv=0)

Answer 10

here it asks me the open interval on which f is increasing what should I put? (-inf,-1/2)U(-1/2,int) ?

Answer 11

yes, very nicely done

Answer 12

and decreasing interval*

Answer 13

empty set

Answer 14

like I just say "it's not decreasing anywhere" ?

Answer 15

yes

Answer 16

Alright, thank you!

Answer 17

welcome