Question

Will medal!

Give intervals where f is concave up and the intervals where f is concave down. Identify all inflection points. F(x)= 2x^6-3x^5-10x^4

I have my F'' being 60x^4-60x^3-120x^2

Answer 1

You can factor out an x^2, then set it equal to zero to find critical values.

Answer 2

Don't you need a denominator? To find where it doesn't exist?

Answer 3

After factoring out 60x^2, you are left with\[60x^2(x^2-x-2)=0\]
Then solve for x, to find concavity of f(x) using f''(x).

Answer 4

Polynomials exist everywhere.

Use the same reasoning as the last question. Find critical values, make a number line.

Concave up means f'' is positive, concave down means f'' is negative.

Inflection points are where the f'' changes sign.

Answer 5

-1 and 2?

Answer 6

You missed one.

Answer 7

-1 and 2 are correct for setting x^2-x-2=0
60x^2=0
divide both sides by 60, x^2=0
sqrt both sides, x=0

Answer 8

I used
F''(-2)
F''(-0.5)
F''(1)
F''(3)

And got all increasing

Answer 9

Of concave up

Answer 10

So are there no inflection points?

Answer 11

I'm guessing you made some mistakes.

https://www.google.com/search?q=2x%5E6-3x%5E5-10x%5E4&oq=2x%5E6-3x%5E5-10x%5E4&aqs=chrome..69i57&sourceid=chrome&ie=UTF-8

That's the graph of the function. It pretty clearly has some regions where it's concave down.

Answer 12

So what did I do wrong?

Answer 13

Probably these
F''(-2)
F''(-0.5)
F''(1)
F''(3)

Answer 14

^Plugging them into f'' is correct.

Answer 15

F'' is the 60x^2-(x^2-x-2) right?

Answer 16

Not quite... 60x^2(x^2-x-2)

Answer 17

Starting with F(x)= 2x^6-3x^5-10x^4, you'll need to find both the first and the second derivatives to answer these questions completely.
I'd suggest you show your work (including intermediate results).

Answer 18

The first derivative is 12x^5-15x^4-40x^3 and the second derivative is 60x^4-60x^3-120x^2

Answer 19

If you used f''(x) = 60x^2-(x^2-x-2) then that would explain why they were all positive.

Answer 20

I did, so what would it be?

Answer 21

It's not 60x^2-(x^2-x-2), it's 60x^2(x^2-x-2)

Answer 22

Okay, this time I got

F''(-2) -
F''(-.5) -
F''(1) -
F''(3) +

Answer 23

Oh I see! I was doing my math wrong in the calculator!

So the inflection points are -1 and 2?

Answer 24

Yep.

Answer 25

Oh okay thank you! I have one more that's kind of similar but I'm going to try and work it myself.

Thank you for all the help and sorry for being all confusing!

Answer 26

You're welcome.

Answer 27

@agent0smith

I'm sorry! One more quick question! I can only find one point in a second derivative test and can't find another

Answer 28

Why do you need more than one point?

Answer 29

Can you only do it with one?

Answer 30

Do it with only one

Answer 31

Yes