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?
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-> 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?
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Question 1036226: 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?
2) When (1) holds, find all x such that C has points in common with the x-axis.
3) When (1) holds, calculate the area S of the limited region bounded by C and the x-axis Answer by Edwin McCravy(20055) (Show Source):
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
