The set of values of $a$ for which $f (x) = x^{3}$ - $a x^{2} + 48 x + 1$ is increasing for all real values of $x$ is

(- 12, 12)

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(- \infty, - 12)

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(1, \infty)

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(- \infty, \infty)

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Solution

The correct option is B $(- 12, 12)$
$f^{'} (x) = 3 x^{2} - 2 a x + 48$
$f^{'} (x) > 0$ $\forall x ϵ R$
$3 x^{2} - 2 a x + 48 > 0$ $\forall x ϵ R$
$3 (x^{2} - \frac{2 a}{3} x + 16) > 0$
$3 ((x - \frac{2 a}{3})^{2} + 16 - \frac{a^{2}}{9}) > 0$
So $16 - \frac{a^{2}}{9} > 0$
$a ϵ (- 12, 12)$

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