Question

Understanding how to find intervals, where functions
1)increase and decrease using 1st Derivative Test.
2)concave up and concave down using 2nd Derivative Test.

Question is;
\[f(x)=\sqrt[3]{x+2}\]

Answer 1

I know where F'(x)>0 then increase and f'(x)<0decrease.
similarly
f''(x)>0 then concave up and f''(x)<0 then concave down.

But is there any specific procedure to determine these properties?

Answer 2

you don't really need calc for the first one.
\[f(x)=\sqrt[3]{x+2}\] looks just like
\[\sqrt[3]{x}\] shifted to the left two units. it is always increasing

Answer 3

the bigger the number, the larger the cubed root

Answer 4

ok. if x is negative then?

Answer 5

\[a<b\implies \sqrt[3]{a}<\sqrt[3]{b}\] i

Answer 6

I know what are you saying. But there is also a specific technique to find increasing and decreasing function by differentiation.

Answer 7

step a, find the places where the derivaite (the slope of any point on the curve) is zero

Answer 8

at x=-2

Answer 9

step 2, pick a point inbetween zeros to test for sign

Answer 10

when sign of the derivative is negative; you are decreasing; when its positive you are increasing; when its zero you are neither increasing nor decreasing

Answer 11

But this function will never be decreasing.

Answer 12

which is why its always increasing except at f'(x) = 0

Answer 13

the same idea with the second derivative to determine cave up or cave down

Answer 14

\[\LARGE f(x)=\frac{1}{3(x+2)^{\frac{2}{3}}}\]

It will never show - sign.

Answer 15

find all critical values; zeros AND undefineds

Answer 16

Sorry it is f'(x)

Answer 17

at x=-2 we are undefined; so test both sides of x=-2

Answer 18

and your right, its never goes negative so that tells us?

Answer 19

One thing I want to ask here. Do for both f '(x) and f ''(x) we have to find critical points?

Answer 20

It tells us that it will always be increasing function.

Answer 21

yes, the critical points partition the interval into testable portions

Answer 22

(-inf , +inf) increasing interval.

Answer 23

good, and since it never goes zero we can determine that x=-2 is also an increasing spot

Answer 24

or do we leave that undefined?

Answer 25

the slope is vertical at x=-2 ...

Answer 26

i forget what do do with x=-2 :) it might need to be opened up

Answer 27

\[\LARGE f''(x)=\frac{-2}{4(x+2)^\frac{5}{3}}\]

It will be negative if x>-2 and positive if x<-2
So concave up interval is (-inf, -2)
And concave down interval is (-2, +inf)

Am I right?

Answer 28

Sorry it is 9 in the denominator not 4

Answer 29

http://www.wolframalpha.com/input/?i=derivative+cbrt%28x%2B2%29

this is showing something wierd to me

Answer 30

Yes it is first derivative. and I have showed that is 2nd derivative.

Answer 31

why is the wolf showing a negative portion on the left of x=-2?

Answer 32

http://www.wolframalpha.com/input/?i=y%3Dcbrt%28x%2B2%29

and this is wacky too; its not showing a translation if cbrt(x)

Answer 33

Firstly I want to tell you one thing which is more important. I have never used any type of graphing utility or any calculator in my life. So I don't know how wolfram alpha works and what it shows. But I believe upon my calculations by hand and mind. :)

Answer 34

oh i trust youre correct since I know what cbrt(x+2) is spose to look like.

but the wolf is usually a good dbl chk to it

Answer 35

I just cant remember what to do at x=-2 with an undefined slope; if we include it in the interval or not

Answer 36

id say exclude it, since a vertical slope is neither inc or dec ....

Answer 37

Now One thing is clear to me. I have learned how to check increament and decrement of the function. But want to know about concavity.

Answer 38

concavity is the same thing; but with f''

Answer 39

Howard Anton Calculus says to include that part i.e. -2 in our case. But some where I have seen not to include critical point. I just got confused :(

Answer 40

id say exclude it; vertical and horizontals dont seem to be defined as inc or dec

Answer 41

but at any rate, use the material that you are using to define it :)

Answer 42

As for f ' (x) we have find critical point. Do for f '' (x) we have to find critical point separately in order to test for sign?

Can you get what I am trying to say?

Answer 43

yes, regards f'' as a separate beast to play with

Answer 44

if it happens to have the same critical points, so be it; but never assume it :)

Answer 45

Yeah i am using howard anton calculus. famous calculus in the world. so I should include critical point. ok i got you.. :) Yes. I got what you have suggested to me.

Answer 46

Now one last thing. what about inflection point?

Answer 47

inflection is where we change concavity; the critical points that have opposite sides to them are inflection; usually where f''(x) = 0 or is undefined need to be considered

Answer 48

I know where curve changes its concavity that is inflection point. but which critical point we have to take as a inflection point?

That one which we have find for f ' (x) or that one which we have find for f '' (x)?

Answer 49

What do you mean by this "that have opposite sides to them are inflection"

Answer 50

where c is a critical point

Answer 51

OK. I have got all that. Thanks once again. :) Be happy always. :)

Answer 52

good luck :)