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The condition for $y = a x^{4} + b x^{3} + c x^{2} + d x + e$ to
have points of inflection is

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$b^{2} - 4 a c > 0$

$3 b^{2} - 8 a c = 0$

$3 b^{2} - 8 a c > 0$

$3 b^{2} - 8 a c < 0$

answer is C.

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Detailed Solution

$y^{'} = 4 a x^{3} + 3 b x^{2} + 2 c x + d$

$y^{''} = 12 a x^{2} + 6 b x + 2 c, y^{''} = 0$ should have real
distinct roots. This is true if $36 b^{2} - 96 a c > 0$

i.e. $3 b^{2} - 8 a c > 0$

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