#### Question

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

(a) \[- 3, - \frac{1}{3}\]

(b) \[- 3, \frac{1}{3}\]

(c) \[3, - \frac{1}{3}\]

(d) \[3, \frac{1}{3}\]

#### Solution

(d) \[3, \frac{1}{3}\]

\[\text { Let }y = x^3 - 5 x^2 + 5x + 8\]

\[ \Rightarrow \frac{dy}{dt} = \left( 3 x^2 - 10x + 5 \right)\frac{dx}{dt}\]

\[\text { According to the question },\]

\[ \Rightarrow 2\frac{dx}{dt} = \left( 3 x^2 - 10x + 5 \right)\frac{dx}{dt}\]

\[ \Rightarrow 3 x^2 - 10x + 5 = 2\]

\[ \Rightarrow 3 x^2 - 10x + 3 = 0\]

\[ \Rightarrow 3 x^2 - 9x - x + 3 = 0\]

\[ \Rightarrow 3x\left( x - 3 \right) - 1\left( x - 3 \right) = 0\]

\[ \Rightarrow \left( x - 3 \right) = 0 \ or \ \left( 3x - 1 \right) = 0\]

\[ \Rightarrow x = 3 \ \text {or} \ x = \frac{1}{3}\]