(a) Use the Constant Difference Theorem (4.8.3) to show that if f ′ x = g ′ x for all x in − ∞ , + ∞ , and if f x 0 − g x 0 = c at some point x 0 , then f x − g x = c for all x in − ∞ , + ∞ . (b) Use the result in part (a) to show that the function h x = x − 1 3 − x 2 + 3 x − 3 is constant for all x in − ∞ , + ∞ , and find the constant. (c) Check the result in part (b) by multiplying out and simplifying the formula for h x .
(a) Use the Constant Difference Theorem (4.8.3) to show that if f ′ x = g ′ x for all x in − ∞ , + ∞ , and if f x 0 − g x 0 = c at some point x 0 , then f x − g x = c for all x in − ∞ , + ∞ . (b) Use the result in part (a) to show that the function h x = x − 1 3 − x 2 + 3 x − 3 is constant for all x in − ∞ , + ∞ , and find the constant. (c) Check the result in part (b) by multiplying out and simplifying the formula for h x .
(a) Use the Constant Difference Theorem (4.8.3) to show that if
f
′
x
=
g
′
x
for all
x
in
−
∞
,
+
∞
,
and if
f
x
0
−
g
x
0
=
c
at some point
x
0
,
then
f
x
−
g
x
=
c
for all
x
in
−
∞
,
+
∞
.
(b) Use the result in part (a) to show that the function
h
x
=
x
−
1
3
−
x
2
+
3
x
−
3
is constant for all
x
in
−
∞
,
+
∞
,
and find the constant.
(c) Check the result in part (b) by multiplying out and simplifying the formula for
h
x
.
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