Question

does f''(x) indicate concavity of f(x)?

Answer 1

f''(x) is the rate of change of the deravitive, so if f''(x) is positive then the derivative is increasing. if the deravitive is increasing then the slope is going from negative to positive.

Answer 2

I'm trying to find the concavity of f(x). How do I find concavity?

Answer 3

concavity is the shape of the derivative.

Answer 4

a positive second deravitive indicates concave up while a negative second deravitive is concave down

Answer 5

Is there a way to show this without plotting a graph? I'm supposed to show, mathematically, my concavity then graph it to see if I am right.

Answer 6

with F''(x) = 0 as your "inflection points" IE. the points where the concavity changes

Answer 7

Is DNE for x also an inflection point?

Answer 8

Do you understand that the second deravitive is the rate of change of the 1st?

Answer 9

Yes, I do.

Answer 10

I wouldnt say DNE is an inflection point because that is usually where the graph tends towards infinity ( could be a jump or removable discontinuity too)

Answer 11

so if the 2nd deravitve is positive then the graph is "swooping" out a bowl. and "swooping" a hill if its negative.

Answer 12

You show it matematically by saying y'' > 0 so f(x) is concave up or vise versa

Answer 13

i think about it with my hand, imaging a swooping motion is change from a negative slope to pos or vise versa

Answer 14

Well, I understand the way it looks. I just don't know how to show it without having to graph it. If I gave you my problem and my answers, would you be able to check my answers?

Answer 15

I have f(x)=ln(x^4+27)

Answer 16

I am asked to find the intervals of increase or decrease

Answer 17

to find the local max and min

Answer 18

to find the intervals of concavity and the inflection points

Answer 19

and to sketch a graph based on my findings

Answer 20

so my f'(x) is 4/x and my f''(x) is -4/x^2

Answer 21

ok this is the first and secont deravitive tests, yea i can check your work, you will be using inequalities to show the shape of the graph analytically

Answer 22

setting both equal to zero, all I get is x DNE

Answer 23

everything bet. neg infinity and zero for f'(x) is negative and positive bet. 0 and infinity

Answer 24

for f''(x) everything is negative

Answer 25

because of my values for f'(x), my values are decreasing from -infinity to zero, and increasing for 0 to infinity

Answer 26

I have no local max because there is only one change of direction at zero (b/c it DNE, there is no inflection point?) and zero is my local min, which DNE?

Answer 27

So no inflection points and based on my f''(x) intervals, I have concave down over the entire graph (of f'(x))?

Answer 28

And that is my answer.

Answer 29

your first deravitve is wrong, (d/dx)(ln(x^4+27)= (4x^3)/(x^4+27)

Answer 30

Ugh, that's what wolfram alpha said, I thought ln(a+b)= ln(a) + ln(b)?

Answer 31

chain rule?

Answer 32

ln(ab) = ln(a) + ln(b)

Answer 33

oy! Thank you. So this is dy/dxln(u) * du/dx (u)?

Answer 34

think of it like exponents, its the exponent rules only reversed

Answer 35

so 1/9x^4+27) X (4x^3)?

Answer 36

sorry, not 9 but (

Answer 37

1/(x^4+27) X (4x^3) = 4x^3/x^4+27

Answer 38

27 is a constant

Answer 39

u = (x^4+27) so that ln(u)'= 1/u du/dx => 1/(x^4+27)*(4x^3) => (4x^3)/(x^4+27)

Answer 40

sorry, nevermind

Answer 41

got it!

Answer 42

yes! you have it right

Answer 43

thank you kindly

Answer 44

okay, so I think I am on my way, thank you very, very much for your help!

Answer 45

so moving on, where the denominator equals zero will give you your asymptotes and where the numerator = zero is your critical numbers (or points of inflection for your second deravitive)

Answer 46

okay

Answer 47

NP, good luck, these questions are rough and tedious

Answer 48

thanks for that, I am starting calc II, but it's been a year since I took calc I, so I have no practice and have forgotten all of this (one doesn't exactly use this sort of life everyday).

Answer 49

what are you talking about, if i got 10 days without showering the rate of growth of bacteria = ln(x) .... lol ok your right, good luck