Algebra Equations Worksheets With Answer Keys Page 8
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20Chapter 5
8
−
x
+
x
e
e
=
=
4. Let
f x
( )
cosh( )
x
.
2
(a) At what x-values does f have relative extrema?
( )
′ −
1
1
<
f
1
0
(
)
(
)
( )
( )
′
−
′
−
x
x
=
x
−
x
=
−
+1
f
x
e
e
f
x
e
e
2
2
( )
′
=
f
0
0
=
+1
x
0
1
(
)
( )
=
x
−
−
x
0
e
e
′
>
f
1
0
+1 Justify extrema
2
−
=
x
−
x
0
e
e
A relative minimum occurs at
1
x = .
0
x
=
e
x
e
2
x
=
e
1
=
2
x
ln1
=
2
x
0
=
x
0
1
∫
(b) Find the exact value for
.
f x dx
( )
−
1
1
1
1
(
)
(
)
−
x
−
x
=
x
−
x
∫
=
−
F x
( )
e
e
+1
F x
( )
e
e
=
−
−
f x dx
( )
F
(1)
F
( 1)
2
2
−
1
+1 Limits
1
(
)
−
=
1
−
1
= −
−
1
F
(1)
e
e
e e
−
1
−
+1
e e
2
1
(
)
−
− =
1
−
1
F
( 1)
e
e
2
( )
( )
(c) Show that f does not have any inflection points.
′′
=
+1
f
x
f x
( )
′′
>
+ 1
f
x
0
everywhere
( )
( )
′′
=
f
x
f x
+1 f concave up everywhere
1
(
)
x
−
x
x
−
x
e >
>
=
+
>
Since
0
and
e
0
for all values of x,
f x
( )
e
e
0
2
( )
′′
⇒
>
for all values of x. Thus
f
x
0
f
concave up everywhere
.
Therefore, f does not have any inflection points.
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