Find a,b, c and d in this function when given the inflection point and a local minimum point?

The function is

f(x)=ax^3+bx^2+cx+d

The local minimum is (3,3) and the inflection point is (2,5)

I really appreciate some help :)

1 Answer
Apr 24, 2018

f(x)=ax^3+bx^2+cx+d

In this function there are 4 unknowns , so 4 equations are required to solve for a, b , c, d

Explanation:

f(x)=ax^3+bx^2+cx+d

rArr f'(x)=3ax^2+2bx+c

And ;

f''(x)=6ax+2b

At Local mimima f'(x)=0 :-

rArr3ax^2+2bx+c=0 ...........where x=3

rArr27a+6b+c=0..........................................................(1)

Also f(3)=3 :-

rArr 3=27a+9b+3c+d...............................................(2)

And f(2)=5 :-

rArr 5=8a+4b+2c+d..................................................(3)

At Inflection point f''(x)=0 :-

rArr 6ax+2b=0....................where x=2

rArr 12a+2b=0

rArr6a+b=0........................................................................(4)

On Solving equations (1),(2),(3),(4) We finally get :-

a=1

b=-6

c=9

d=3

Thus the given function is :

f(x)=x^3-6x^2+9x+3