Math, asked by sharantt03, 4 months ago

f (x) = 1 + cos2 – a increases if a is​

Answers

Answered by chrisjames55
0

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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