SOLUTION: Let C be the plane curve y=f(x) defined by the cubic function f(x)= x^3 - 4x^2 + ax + b with a, b real constants... 1. When C is tangent to the x-axis at x=3, what are a and b?

Algebra ->  Test -> SOLUTION: Let C be the plane curve y=f(x) defined by the cubic function f(x)= x^3 - 4x^2 + ax + b with a, b real constants... 1. When C is tangent to the x-axis at x=3, what are a and b?      Log On


   

Question 1037481: Let C be the plane curve y=f(x) defined by the cubic function f(x)= x^3 - 4x^2 + ax + b with a, b real constants...
1. When C is tangent to the x-axis at x=3, what are a and b?
2. When (1) holds, find all x such that C has points in common with th x-axis.
3. When (1) holds, calculate the area S of the limited region bounded by C and the x-axis.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
1. f%28x%29=+x%5E3+-+4x%5E2+%2B+ax+%2B+b being tangent to the x-axis at x = 3 means two things:
f%283%29=+3%5E3+-+4%2A3%5E2+%2B+3a+%2B+b+=+-9%2B3a%2Bb+=+0, and
f'{3) = 3%2A3%5E2+-+8%2A3+%2B+a+=+0.
==> 3a+b = 9 and 3+a=0
==> a = -3 and -9 + b = 9 ==> b = 18.
==>f%28x%29=+x%5E3+-+4x%5E2+-+3x+%2B+18
2. f%28x%29=+x%5E3+-+4x%5E2+-+3x+%2B+18+=+%28x%2B2%29%28x-3%29%5E2. Therefore all x-values such that C has points in common with th x-axis are x = -2, 3, 3. (3 is a double root.)
3.