find the derivation of function xx + (cot x) x
Answers
Answer:
Given y=f(x)=
x+cotx
xcotx
∴y=
x+
tanx
1
tanx
x
∴y=
xtanx+1
x
For deciding whether function is maximum or minimum between (0,
2
π
), differentiate function w.r.t. x.
∴
dx
dy
=
dx
d
(
xtanx+1
x
)
∴
dx
dy
=
(xtanx+1)
2
(xtanx+1)
dx
d
(x)−x
dx
d
(xtanx+1)
∴
dx
dy
=
(xtanx+1)
2
(xtanx+1)(1)−x(xsec
2
x+tanx)
∴
dx
dy
=
(xtanx+1)
2
xtanx+1−x
2
sec
2
x−xtanx
∴
dx
dy
=
(xtanx+1)
2
1−x
2
sec
2
x
∴f
′
(x)=
(xtanx+1)
2
1−x
2
sec
2
x
Now, f(x) is continuous in (0,
2
π
)
At x=0, f(0)=
(0+1)
2
1−0
∴f(0)=
1
1
=1>0
Thus, function will be increasing at x=0
At x=
2
π
, f
′
(x)=
(∞+1)
2
1−∞
∴f(
2
π
)<0
Thus, function is decreasing at x=
2
π
Thus, we can conclude that function is increasing initially and then decreasing.
Thus, function will attend local maxima in the interval (0,
2
π
)