Relative Extrema: Find a,b,c,d for f(x)

[jhodzzz]

Homework Statement

I have to find a, b, c, and d such that the function defined by :
f(x) = ax³+bx²+cx+d
will have a relative extrema at points (1,2) and (2,3).

The Attempt at a Solution

From the given critical points, I am able to know that when x=1 or x=2, f'(x)=3ax²+2bx+c should be equal to zero.

Therefore f'(x) should have factors like (x-1) and (x-2) or in simplified form x²-3x+2. Now my problem is that how should I relate the two equations of f'(x) for me to be able to solve for the said unknowns?

 

[Gib Z]

Well you know f'(x) = 3ax^2 + 2bx + c = 0 when x=1 and x=2, So substitute that in. You will get two equations which you can view as linear equations in the variables a, b, c, d, with the coefficient of d being zero.

What else do you need to solve this system and how can you get that information?

 

[jhodzzz]

Well you know f'(x) = 3ax^2 + 2bx + c = 0 when x=1 and x=2, So substitute that in. You will get two equations which you can view as linear equations in the variables a, b, c, d, with the coefficient of d being zero.

What else do you need to solve this system and how can you get that information?

>>> after getting the two equations: 3a + 2b + c and 12a + 4b + c, what will I do... I still could not find a way to get the values of a, b, c, and d... after eliminating c, I only get 9a + 2b giving two variables unknown still.. help.. :(

 

[Gib Z]

You can get two more equations. Look at all your information again!