Application of DerivativeHard
Question
Let f(x) = x3 + ax2 + bx + 5 sin2 x be an increasing function in the set of real numbers R. Then a & b satisfy the condition :
Options
A.a2 - 3b - 15 > 0
B.a2 - 3b + 15 ≤ 0
C.a2 + 3b - 15 < 0
D.a > 0 & b > 0
Solution
f′(x) = 3x2 + 2ax + b + 5 sin 2x ≥ 0 ∀ x ∈ R
⇒ sin 2x ≥ - 1
⇒ f′(x) ≥ 3x2 + 2ax + b - 5 ∀ x ∈ R
⇒ 4a2 - 4 .3 . (b - 5) ≤ 0
⇒ a2 - 3b + 15 ≤ 0
⇒ sin 2x ≥ - 1
⇒ f′(x) ≥ 3x2 + 2ax + b - 5 ∀ x ∈ R
⇒ 4a2 - 4 .3 . (b - 5) ≤ 0
⇒ a2 - 3b + 15 ≤ 0
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