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If $f (x) = x^{3} + a x^{2} + b x + 5 \sin^{2} x$ is an increasing function on IR, then

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$a^{2} - 3 b + 15 > 0$

$a^{2} - 3 b - 15 < 0$

$a^{2} - 3 b - 15 > 0$

$a^{2} + 3 b + 15 > 0$

answer is A.

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Detailed Solution

$f (x) = x^{3} + a x^{2} + b x + 5 \sin^{2} x$

$f' (x) = 3 x^{2} + 2 a x + b + 5 \sin 2 x > 0$

$∴ b^{2} - 4 a c < 0$

$\Rightarrow {(2 a)}^{2} - 4 (3) (b + 5 \sin 2 x) < r$

$a^{2} - 3 (b + 5 \sin 2 x) < 0$

$∴ a^{2} - 3 b - 15 < 0$ ( Max. $\sin 2 x = 1$ )

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