The curve $y = a x^{3} + b x^{2} + c x + d$ has a point on inflexion at $x = 1$ then;

a + b = 0

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a + 3 b = 0

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3 a + b = 0

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3 a + b = 1

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Solution

The correct option is B $3 a + b = 0$
$y = a x^{3} + b x^{2} + c x + d$
$\Rightarrow y^{'} = 3 a x^{2} + 2 b x + c$ , differentiate given curve
$\Rightarrow y^{''} = 6 a x + 2 b$
So for $x = 1$ to be inflection point
$y^{''} |_{x = 1} = 0$
$\Rightarrow 6 a (1) + 2 b = 0 \Rightarrow 3 a + b = 0$

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