For what values of x is the rate of increase of $x^{3} - 5 x^{2} + 5 x + 8$ is twice 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 $y = x^{3} - 5 x^{2} + 5 x + 8$ . Then,
$\frac{d y}{d x} = (3 x^{2} - 10 x + 5) \frac{d x}{d t} W h e n \frac{d y}{d t} = 2 \frac{d x}{d t}, w e h a v e (3 x^{2} - 10 x + 5) \frac{d x}{d t} = 2 \frac{d x}{d t} \Rightarrow 3 x^{2} - 10 x + 3 = 0 \Rightarrow (3 x - 1) (x - 3) = 0 \Rightarrow x = 3, \frac{1}{3}$ .
Hence (d) is the correct answer.

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

x

x^{3} - 5 x^{2} + 5 x + 8

x

y = 6 x - x^{3}

x

5

x = 3

x = 3

\frac{x^{3} - 5 x^{2} + 7 x - 3}{x^{3} - x^{2} - 5 x - 3}

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