Question

Intervals of concavity, can anyone explain how to do this?

Answer 1

Do you have a function we can work with?

Answer 2

YES, omg. didnt think anyone would reply.
one second.

Answer 3

Basically you need to find when \[
f''(x)=0
\]

Answer 4

right. i got halfway there i think.
\[f(x)=2\sqrt{x}-x \]

Answer 5

That is the inflection points. Concavity can only change at inflection points.

Answer 6

is the equation, first derivative is 1/radical(x)-1

Answer 7

Now second derivative?

Answer 8

i got x=1 but im not sure thats correct.

Answer 9

Second derivative is?

Answer 10

\[\frac{ 1 }{-2x ^{3/2} }\]

Answer 11

Okay so does it have any roots?

Answer 12

Um. I dont think so. no.

Answer 13

So the concavity never changes. Is it always concave up or always concave down?

Answer 14

what exactly are roots again?

Answer 15

Roots of \(f(x)\) is just any \(x\) when \(f(x)=0\)

Answer 16

so you'd set the second derivative equal to 0?

Answer 17

Yes.

Answer 18

technically couldn't you solve for x and get a number?

Answer 19

Try it out: =)

Answer 20

I cant doo it. Thats annoying. you end up with -1/x^3. I want to make it something. o.o

Answer 21

Okay well think about it this way \[
\frac{f(x)}{g(x)} = 0
\]Multiply both sides by \(g(x)\): \[

f(x)=0 \cdot g(x)=0
\]So the roots of \(f(x)/g(x)\) are just the roots of \(f(x)\)

Answer 22

In this case \(f(x)=1\).
We know \(1\) has no roots because... \(1\ne 0\).

Answer 23

Thus \[
\
\frac{f(x)}{g(x)} =\frac{ 1 }{-2x ^{3/2} }=0
\]Has no solutions... there are no roots.

Answer 24

So the function is always concave up or always concave down

Answer 25

@skay I don't have all night. Say something.

Answer 26

Okay. sorry

Answer 27

Just thinking it through.

Answer 28

umm. its concave down.

Answer 29

How did you figure that out?

Answer 30

I graphed it with my calculator :(

Answer 31

All you got to do is put in a test point, like \(x=1\).

Answer 32

0.35?

Answer 33

You need to try harder and ask questions. I'm running out of time.

Answer 34

ummm in the second derivative?

Answer 35

Yes.

Answer 36

okay so what do i do with 0.35?

Answer 37

That isn't what you get when you put \(1\) in the second derivative.

Answer 38

i get negative 1/2.8284...

Answer 39

wait, -1/2?

Answer 40

Okay is that above zero or below zero?

Answer 41

below

Answer 42

so x<0

Answer 43

No \(f''(x)<0\)
Which means that \(f(x)\) is concave down.

Answer 44

Right. according to the rules. Ok.

Answer 45

thanks!

Answer 46

What would my intervals be if there is no roots?

Answer 47

\[
(-\infty , \infty)
\]

Answer 48

Thought so, thanks!