For what values of $x$ is the rate of increase of $x^{3} - 5 x^{2} + 5 x + 8$ is twice the rate of increase of $x$ ?

- 3, - \frac{1}{3}

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- 3, \frac{1}{3}

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3, - \frac{1}{3}

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3, \frac{1}{3}

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Solution

The correct option is D $3, \frac{1}{3}$
Let $y = x^{3} - 5 x^{2} + 5 x + 8$

$\frac{d y}{d t} = (3 x^{2} - 10 x + 5) \frac{d x}{d t}$
Given, $\frac{d y}{d t} = 2 \frac{d x}{d t}$
$\Rightarrow (3 x^{2} - 10 x + 5) \frac{d x}{d t} = 2 \frac{d x}{d t}$

$\Rightarrow (3 x^{2} - 10 x + 3) \frac{d x}{d t} = 0$
$\Rightarrow 3 x^{2} - 10 x + 3 = 0$

$\Rightarrow 3 x^{2} - 9 x - x + 3 = 0$

$\Rightarrow 3 x (x - 3) - 1 (x - 3) = 0$

$\Rightarrow (x - 3) (3 x - 1) = 0$
$\Rightarrow x = 3, \frac{1}{3}$

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