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
Kalit Gautam
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#

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