Question

f(x) = x^3 + ax^2 + bx + c
Find the consonants a,b, and c so that the graph will increase to the point (-3,18), decrease to the point (1, -14), and then continue increasing.

Answer 1

So we have critical numbers do you agree?
What are the 2 critical numbers?

Answer 2

notisaiah?

Answer 3

The critical points are found by taking the first derivative.

Answer 4

Great!

Answer 5

Now since these are critical numbers and this is a poly that would mean f'(-3)=0
and f'(1)=0

Answer 6

okay

Answer 7

So now phi said you need to find the first derivative and he is correct

Answer 8

Could you tell me what you get for f'(x)?

Answer 9

Wait, so I find the derivative of the function, then i plug in -3 and 1 ?

Answer 10

f`(x) = 3x^2 + 2ax + b

Answer 11

Great!
So Evaluate f' at x=-3 and evaluate f' at x=1
Do we not have more information?

Answer 12

I'm not seeing how to find c right now
but this will give you two linear equations that you can solve for a and b

Answer 13

No, that is it

I get f`(-3) = 27 - 6a + b and f`(1) = 3 + 2a + b

Answer 14

And both of those are =0

Answer 15

27-6a+b=0
3+2a+b=0

Answer 16

yes.

Answer 17

This was the set of linear equations I was talking about

Answer 18

right.

Answer 19

Oh I know how to get c I'm an idiot
I wasn't thinking that far ahead but we are given two points on f

Answer 20

So once you find a and b from there
We will go back to the original function f
and do f(-3)=18 and f(1)=-14 to find c

Answer 21

or*

Answer 22

could you elaborate please

Answer 23

Which part?
Finding a and b?
Or using f(1)=-14 to find c?

Answer 24

What I have right now are the two functions
I get f`(-3) = 27 - 6a + b + c
and f`(1) = 3 + 2a + b + c

Answer 25

oops, without the c
okay let me find the values for a and b

Answer 26

Remember those are both equal to 0

Answer 27

remember these points are at a max or a min, so the slope is 0

Answer 28

okay so I have the two functions f`(-3) = 27 - 6a + b and f`(1) = 3 + 2a + b
where b = -9 , a = 3

Answer 29

now use the original equation with your a,b values

Answer 30

So now I have f`(x) = x^3 + 3x^2 - 9x + c
How do I find the c value

Answer 31

It's f(x) not f'(x). f(x) is another name for the 2nd value (y value) in an (x,y) pair that satisfies the equation. You have 2 points. Choose one of them.

Answer 32

I see. thank you !