f(x)=sinx-cosx find maximum point
Answers
Answer:
Please see the explanation.
Explanation:
The x coordinates of extrema can be found by, computing the first derivative, setting that equal to zero, and then solving for x:
Compute the first derivative:
f
'
(
x
)
=
cos
(
x
)
−
sin
(
x
)
Set equal to zero:
0
=
cos
(
x
)
−
sin
(
x
)
Solve for x:
cos
(
x
)
=
sin
(
x
)
1
=
sin
(
x
)
cos
(
x
)
1
=
tan
(
x
)
x
=
tan
−
1
(
1
)
x
=
π
4
Because the tangent function has a period of
π
, the value, 1, repeats at every integer multiple of
π
:
x
=
π
4
+
n
π
where
n
=
...
,
−
3
,
−
2
,
−
1
,
0
,
1
,
2
,
3
,
...
To determine whether this is a maximum, perform the second derivative test, using one of the values:
f
'
'
(
x
)
=
−
cos
(
x
)
−
sin
(
x
)
Evaluate at
π 4 f ' ( π 4 )=− cos (π 4 ) − sin ( π 4 ) = − √ 2
The value is a negative, therefore, we have found a maximum