f (x) = 1 + cos2 – a increases if a is
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Step-by-step explanation:
f(x)=sin2x−8(a+1)sinx+(4a
2
−8a−14)x
⇒f
′
(x)=2cos2x−8(a+1)cosx+(4a
2
+8a−14)
−1≤cos2x.cosx≤1
if a≥−1,
⇒−2−8(a+1)+8a+4a
2
−14≤f
′
(x)≤2+8(a+1)+4a
2
+8a+4
4a
2
−24≤f
′
(x)≤4a
2
+16a−4
4(a
2
−6)≤f
′
(x)≤4(a
2
+4a−1)
a>
6
0≥4(a
2
−6)≤f
′
(x)
If a<1,
⇒−2+8(a+1)+4a
2
+8a−14≤f
′
(x)≤2−2(a+1)+4a
2
+8a+4
4a
2
+16a−8≤f
′
(x)≤4a
2
−20
4(a
2
+4a−2)≤f
′
(x)≤4(a
2
−20)
If a≤(−2−
6
)=1 0<4(a
2
+4a−2)≤f
′
(x)
∴∀a∈(−∞,−2−
6
)0(
6
,∞), f
′
(x)≥0
⇒f(x) increases ∀x∈R& has no critical points.
∴ at (−∞,−
5
−2) is not possible.
at (1,∞) is not possible
& a∈(
7
,∞) is possible.
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