The condition that $f (x) = a x^{3} + b x^{2} + c x + d$ has no extreme value is

b^{2} > 3 a c

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b^{2} = 4 a c

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b^{2} = 3 a c

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b^{2} < 3 a c

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Solution

The correct option is D $b^{2} < 3 a c$
$f (x) = a x^{3} + b x^{2} + c x + d$
$\Rightarrow f^{'} (x) = 3 a x^{2} + 2 b x + c$
For no extrema, $f$ should have zero turning points, i.e. above quadratic have no real roots.
$\Rightarrow (2 b)^{2} - 4 \cdot 3 a \cdot c < 0$
$\Rightarrow b^{2} < 3 a c$

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