If the set of all values of the parameter 'a' for which the function
f(x) = sin2x – 8(a + 1) sin x + (4a 2 + 8a – 14) x increases for all x ∈ R and has no critical points for all x ∈ R, is (– ∞ , –m – 
) ∪ ( 
, ∞ ) then (m 2 + n 2 ) is (where m, n are prime numbers) :
Text Solution
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(29)
Sol. f ′ (x) = 4 
Let φ (t) = t 2 – 2(a + 1)t + a 2 + 2a – 4, –1 ≤ t ≤ 1
It is sufficient to find values of t when φ (t) = 0 has no root in [–1, 1]
D = 20
Case- Ι :
φ (–1) > 0 , a + 1 < –1, D = 20 > 0
⇒ a ∈ 
∪ 
and a < –2
⇒ a ∈ 
Case - ΙΙ :
φ (1) > 0, a + 1 > 1, D = 20 > 0
⇒ a ∈ 
∪ 
and a > 0
⇒ a ∈ 
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