Determine the condition so that the function $f (x) = x^{3} + p x^{2} + q x + r$ is an increasing function for all real x.

p^{2} - 3 q < 0

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p^{2} - 3 q > 0

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q^{2} - 3 p < 0

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q^{2} - 3 p > 0

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Solution

The correct option is A $p^{2} - 3 q < 0$
Given $y = x^{3} + p x^{2} + q x + r$
$∴ \frac{d y}{d x} = 3 x^{2} + 2 p x + q$
Now for $f (x)$ to be increasing function for all x, $f^{'} (x) > 0$
and for this discriminant of $f^{'} (x)$ should be negative
$\Rightarrow D = 4 p^{2} - 12 q < 0$ or $p^{2} - 3 q < 0$

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