Product Rule (Thinking/Inquiry)

[ghostanime2001]

Determine a quadratic funtion $$f(x) = ax^2 + bx + c$$ who graph passes through the point (2,19) and that has a horizontal tangent at (-1,-8).

My attempt at this solution is:
$$f(x) = ax^2 + bx + c$$
$$f'(x) = 2ax + b$$

LOL its not much but i really have absolutely no idea where to go from here :S

 

[gabbagabbahey]

Have you tried substituting the given point and tangent point into the equations for f and f' respectively? What do you get if you do that?

 

[ghostanime2001]

All i know is
$$f(x) = ax^2 + bx + c$$
$$f'(x) = 2ax + b$$
$$f'(-1) = 2a(-1) + b$$
$$f'(-1) = -2a + b$$
$$0 = -2a + b$$ @ x = -1
$$0 = 4a + b$$ @ x = 2

After that i know i have to find 2 variables (a and b) and then finally find c but before i can do that i have to know what to do with those 2 equations which i don't @_@

 

[gabbagabbahey]

If f(x) passes through (2,19) then f(2)=19 , similarly if f has a horizontal tangent at (-1,-8), then f'(-1)=0 and f(-1)=8...this should give you 3 equations and 3 unknowns a,b,c which you can solve for. Can you take it from there?

 

[ghostanime2001]

so u are saying:
$$19=4a+2b+c$$ for f(2)=19
$$0=-2a+b$$ for f'(-1)=0
$$-8=a-b+c$$ for f(-1)=8

That also means I am solving a system of 3 equations correct ? oh crap... i need to review my basics T_T

 

[gabbagabbahey]

so u are saying:
$$19=4a+2b+c$$ for f(2)=19
$$0=-2a+b$$ for f'(-1)=0
$$-8=a-b+c$$ for f(-1)=8

That also means I am solving a system of 3 equations correct ? oh crap... i need to review my basics T_T

Yes, but do you understand why f(2)=19, f'(-1)=0 and f(-1)=-8? And yes, you will need to solve a system of 3 equations to determine a b and c.

 

[ghostanime2001]

YAY ! SOLVED IT

I DONT NEED HELP ANYMORE it was so simple... 3 unknowns 3 equations lol sorry for caps this is so exiciting !

 

[ghostanime2001]

and yes i do know why ... i think if u would so kindly tell me i can verify my thought. Is it because we *neeed* 3 equations and the only ones we know are from those 2 co-ordinates

 

[HallsofIvy]

No, those are not true because you need them.

f(2)=19 is true because you were TOLD that the graph goes through (2, 19).
f'(-1)=0 is true because you were TOLD that the graph has a horizontal tangent at (-1, -8) and a "horizontal tangent" has slope 0.
f(-1)= -8 because in order to have a tangent line at all at (-1, -8), the graph must include the point (-1, -8).

The fact that you need three equations to determine the three coeffiencents is why you use them, not why they are true!

 

[ghostanime2001]

Oh i know that... i had the grand unification theory in my head. I thought it would be explained in a sentence or so. my bad T_T