The condition that $f (x) = x^{3} + a x^{2} + b x + c i s a n$ increasing function for all real values of $x$ is

a^{2} < 12 b

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a^{2} < 3 b

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a^{2} < 4 b

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a^{2} < 16 b

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Solution

The correct option is D $a^{2} < 3 b$
$f^{1} (x) = 3 x^{2} + 2 a x + b$
$f^{1} (x) > 0$
$3 x^{2} + 2 a x + b > 0$
Only If $(2 a)^{2} - 4 \times 3 b < 0$ (Property of quadratic $e q^{n}$ )
$a^{2} < 3 b$

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