(a) Prove that a general cubic polynomial f x = a x 3 + b x 2 + c x + d a ≠ 0 (b) Prove that if a cubic polynomial has three x -intercepts, then the inflection point occurs at the average value of the intercepts. (c) Use the result in part (b) to find the inflection point of the cubic polynomial f x = x 3 − 3 x 2 + 2 x , and check your result by using f ″ to determine where f is concave up and concave down.
(a) Prove that a general cubic polynomial f x = a x 3 + b x 2 + c x + d a ≠ 0 (b) Prove that if a cubic polynomial has three x -intercepts, then the inflection point occurs at the average value of the intercepts. (c) Use the result in part (b) to find the inflection point of the cubic polynomial f x = x 3 − 3 x 2 + 2 x , and check your result by using f ″ to determine where f is concave up and concave down.
(a) Prove that a general cubic polynomial
f
x
=
a
x
3
+
b
x
2
+
c
x
+
d
a
≠
0
(b) Prove that if a cubic polynomial has three x-intercepts, then the inflection point occurs at the average value of the intercepts.
(c) Use the result in part (b) to find the inflection point of the cubic polynomial
f
x
=
x
3
−
3
x
2
+
2
x
,
and check your result by using
f
″
to determine where
f
is concave up and concave down.
2.
Complete steps #A-E below and graph this polynomial:
f(x) = x' + 3x² – 9x – 27
A) Find the zeros. (Show work.)
B) Identify the multiplicity of each zero above.
C) Divide the x-axis and use test values of x to determine signs.
Interval
Test Value
Function Value f(x)
Sign of f(x)
D) Find f(0) =
E) Use all of the information above
to create an accurate sketch of
this function. Draw it in the space
to the right.
->
Consider a general cubic polynomial f(x)=ax3+bx2+cx+d
1. Find the turning points (extreme points). Remember, points have both an x and a y coordinate.
2. Find the inflection point.
The curve represented by r(t) = f(t)i + g(t)j + h(t)k is a line. Are f, g, and h first-degree polynomial functions of t? Explain.
Precalculus: Mathematics for Calculus - 6th Edition
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