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Question: If f(x) = \(\frac{1}{3}\) (x<sup>3</sup> + 3ax<sup>2</sup> + 3bx – cos<sup>2</sup> x), x Î R, is an ...

If f(x) = 13\frac{1}{3} (x3 + 3ax2 + 3bx – cos2 x), x Î R, is an increasing function for all x Î R, then :

A

a2> b – (1/3)

B

a2< b – (1/3)

C

a2> b + (1/3)

D

a2< b + (1/3)

Answer

a2> b + (1/3)

Explanation

Solution

f (x) = 13\frac{1}{3} (x3 + 3ax2 + 3bx–(a2l) is increasing for all x Î R.

f ' (x) = x2 + 2a x + b + sin⁡2x3\frac{\sin 2x}{3}> 0 ∀\forall x Î R.

= (x + a)2 + sin⁡2x3\frac{\sin 2x}{3}> a2 – b ∀\forall x Î R.

̃ a2 – b < – 13\frac{1}{3} Q Min of (x + a)2 = 0

̃ a2 < b – 1/3 Min of sin⁡2x3\frac{\sin 2x}{3}= –1/3