SOLUTION: Hi, I am terrible at systems of equations and I have an applications of derivatives assignment with a few questions, the only one dealing with systems of equations is as follows:

Question 858205: Hi, I am terrible at systems of equations and I have an applications of derivatives assignment with a few questions, the only one dealing with systems of equations is as follows:
For the function f(x)= ax^4 + bx^3 - 4x^2 + 2cx + 14
Determine the values of a, b, and c, such that,
f(-2)=2 , f'(-2)=16 , and f''(-2)=-8

I have tried only messing around with various numbers in my calculator for the past hour to get any number to match with the given values, and I failed
Any help would be much appreciated
Thanks
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
For the function f(x)= ax^4 + bx^3 - 4x^2 + 2cx + 14
Determine the values of a, b, and c, such that,
f(-2)=2 , f'(-2)=16 , and f''(-2)=-8
-------
f(x)= ax^4 + bx^3 - 4x^2 + 2cx + 14
f(-2) = 16a - 8b -16 - 4c + 14 = 2
16a - 8b - 4c = 4
4a - 2b - c = 1 Eqn 1
----
f(x)= ax^4 + bx^3 - 4x^2 + 2cx + 14
f'(x) = 4ax^3 + 3bx^2 - 8x + 2c
f'(-2) = -32a + 12b + 16 + 2c = 16
16a - 6b - c = 0 Eqn 2
----------
f''(x) = 12ax^2 + 6bx - 8
f''(-2) = 48a - 12b - 8 = -8
4a - b = 0 Eqn 3
------
b = 4a
Sub for b in Eqns 1 & 2
---
4a - 2b - c = 1 Eqn 1
4a - 8a - c = 1
4a + c = -1 Eqn A
---
16a - 6b - c = 0 Eqn 2
16a - 24a - c = 0
8a + c = 0
c = -8a
----
Sub for c in Eqn A
4a + c = -1 Eqn A
4a - 8a = -1
a = 1/4
-------
c = -2
b = -1
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