Question

Need help.
Need to find the concavity and inflection point of
e^x/(8+e^x)

Answer 1

have you started anythign yet?

Answer 2

I have the second derivative

Answer 3

what are the two derivatives you got

Answer 4

\[\frac{ 8e ^{x}(8+e ^{x})^{2}-16e ^{2x}(8+e ^{x}) }{(8+e ^{x})^{4} }\]
that's what I have for the second derivative

Answer 5

your second derivative is correct. I just did it

Answer 6

so where do I go from there?

Answer 7

inflections exist where \[f''(x) = 0\]

Answer 8

and there are two types of concavity.
concave upward
concave downward
you need to find them too in your problem

Answer 9

I just don't know how to solve the second derivative=0 to get the inflection points

Answer 10

okay. can you simplify numerator of second derivative.

Answer 11

you need simplify numerator. there are some term that cancels each other.

Answer 12

and numerator means upper part of a fraction

Answer 13

you can only cancel one of the (8+e^x) right?

Answer 14

did you get
\[f''(x) = \frac{64e^x}{(8+e^e)^3}\]

Answer 15

\[f''(x) = \frac{64e^x}{(8+e^x)^3}\]

Answer 16

is it?

Answer 17

ya

Answer 18

okay. To find inflection second derivative should be equal to 0

\[\frac{64e^x}{(8+e^x)^3} = 0\]\[64e^x = 0\]

Answer 19

get it?

Answer 20

yes

Answer 21

if you want a fraction to be equal to zero, you can't forget denominator(down part of fraction)

Answer 22

so this means
\[e^x = 0\]
and this is the graph of e^x. I will draw it, so that you can understand easily.