How to Determine the Coefficients of a Cubic Function with Given Conditions?

[dontdisturbmycircles]

Homework Statement

Find a,b,c, and d such that the cubic f(x)=$$ax^{3}+bx^{2}+cx+d$$satisfies the indicated conditions.

Relative maximum (3,3)
Relative minimum (5,1)
Inflection point (4,2)

Homework Equations

The Attempt at a Solution

I am so lost as to how to do this :/.

Its a polynomial so f ' (x) must = 0 at x=3 and x=5 (can't not exist), and I also know that the derivative of f(x) will be a function of degree 2, which can have at most two roots. Thus the function must be of the form a(x-3)(x-5)=f ' (x), right?

I know that the second derivative is defined for all x (can't have negative exponents, they would become constants before that point). And that f '' (x)=0 at x=4...

I just can't see how to piece it all together. Can someone help me out?

 

[cristo]

Work out the explicit form of the first and second derivatives of f. Then plug in x=3 and x=5 into the first derivative, which you said must be zero. Plug in x=4 into the second derivative, which again, you said must be zero. This will give you 3 equations in 3 unknowns, which you can solve.

 

[dextercioby]

$$f'(x)=3a(x-3)(x-5)\equiv 3ax^2 +2bx +c$$

is a place to start.