the set of all values of f(x)=1/3sin^ 2 x+sinx cosx+cos^ 2x is
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Answer:
f(x)=(a
2
=−3a+2)(cos
2
x/4−sin
2
x/4)+(a−1)x+sin1
⇒f(x)=(a−1)(a−2)cosx/2+(a−1)x+sin1
⇒f
′
(x)=−
2
1
(a−1)(a−2)sin
2
x
+(a−1)
⇒f
′
(x)=(a−1)[1−
2
(a−2)
sin
2
x
]
If f(x) does not possess critical points, then f
′
(x)
=0 for any xϵR
⇒(a−1)[1−
2
(a−2)
sin
2
x
]
=0 for any xϵR
⇒a
=1 and 1−(
2
a−2
)sin
2
x
=0
must not have any solution in R.
⇒a
=1 and sin
2
x
=
a−2
2
is not solvable in R.
⇒a
=1 and
∣
∣
∣
∣
∣
a−2
2
∣
∣
∣
∣
∣
>1 [For a=2,f(x)=x+sin1∴f
′
(x)=1
=0]
⇒a
=1 and ∣a−2∣<2⇒a
=1 and −2<a−2<2
⇒a
=1 and 0<a<4⇒aϵ(0,1)∪(1,4).
Step-by-step explanation:
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