Algebra Equations Worksheets With Answer Keys Page 10
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20Chapter 5
10
=
+
=
−
5. Let
f x
( )
cosh( ) sinh( )
x
x
and ( )
g x
cosh( ) sinh( )
x
x
.
( )
(a) Use calculus to show that
f x is an increasing function.
( )
1
1
′
=
x
(
)
(
)
+1
f x
e
( )
−
−
=
x
+
x
+
x
−
x
f x
e
e
e
e
2
2
( )
′
>
+1
f
x
0
x
=
e
+1 Justify increasing
( )
′
=
x
>
f
x
e
0 for all
x
( )
′
>
Since
f
x
0
for all x , f is increasing.
( )
( ) ( )
=
(b) Define
h x
f x g x
. Determine the critical values of h .
( )
′
2
2
=
=
−
h x
( )
cosh
x
sinh
x
+1
h x
0
( )
(
)(
)
(
)(
)
′
=
−
+1 All values of x are
h x
2 cosh
x
sinh
x
2 sinh
x
cosh
x
critical values
=
0
( )
′
=
Since
h x
0
for all values of x , every x-value is a critical value.
( )
(c) Write the expressions in order from the smallest value to the largest
=
+1
h
0
1
( )
( )
′
′
value:
h
(0),
h
0 ,
f
1
. Show the work that leads to your
( )
h ′
=
+1
0
0
conclusion.
( )
′
=
+1
f
1
e
( )
+1 correct order
=
f
(0) 1
From (a),
h ′
= .
From (b),
0
0
( )
′
= ≈
f
1
e
2.718
=
g
(0) 1
=
h
(0)
f
(0) (0)
g
( )
=
1 1
=
1
( )
( )
( )
′
′
<
<
So
h
0
h
0
f
1
.
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