If x∈(0 ,
π
/2
) then find minimum of the function sec
^2
x+cot^6
x ?
Answers
Answered by
0
A)
f(x)
=
x+cotx
xcotx
=
1+xtanx
x
f
′
(x)=
(1+xtanx)
2
1−x
2
sec
2
x
put
′
f
′
(x)=0
⇒1=x
2
sec
2
xx∈(0,
2
π
)
⇒x=cosxx∈(0,
2
π
)
f
′′
(x)∣
x=cosx
=
x(1+xtanx)
−
2
1
<0
⇒f(x) has local maxima in (0,
2
π
)
f(x)
=
x+cotx
xcotx
=
1+xtanx
x
f
′
(x)=
(1+xtanx)
2
1−x
2
sec
2
x
put
′
f
′
(x)=0
⇒1=x
2
sec
2
xx∈(0,
2
π
)
⇒x=cosxx∈(0,
2
π
)
f
′′
(x)∣
x=cosx
=
x(1+xtanx)
−
2
1
<0
⇒f(x) has local maxima in (0,
2
π
)
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