SOLUTION: Let C be the plane curve y = f(x) defined by the cubic function f(x) = x³ - 4x² + ax + b with a, b real constants 1) When C is tangent to the x-axis at x=3, what are a and b?

f(x) = x³ - 4x² + ax + b

tangent to the x-axis at x=3,

Therefore 3 is a double zero, (zero of even 
multiplicity, which can only be multiplicity 2,
Therefore synthetic division will produce a 
remainder of 0 twice:


  3|1 -4  a     b
   |   3 -3   3a-9  
    1 -1 a-3  3a+b-9 = 0


  3|1 -1 a-3
   |   3  6
    1  2 a+3 = 0

So f(x) factors as 

f(x) = (x-3)²(x+2)

So setting both those remainders = 0, gives
is a system of those two equations in two unknowns:

        a+3 = 0
          a = -3

      a+b-3 = 0
     -3+b-3 = 0
        b-6 = 0
          b = 6

f(x) = x³ - 4x² + ax + b
f(x) = x³ - 4x² - 3x + 18
 and its factorization is:

f(x) = (x-3)²(x+2)

The zeros are 3 and -2.  

The curve C and the x-as axis have points (3,0) and (-2,0)
in common. 



To find the area bounded by C and the x-axis is given by


                       |3
   | 
                       |-2

   

Approximately 52.083

Edwin