Question

For what values of x is the rate of increase of x³ − 5x² + 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) $3,\frac{1}{3}$

Answer 1

(d) $3,\frac{1}{3}$

$$\begin{gathered} \mathrm{Let}y=x^3-5x^2+5x+8 \\ \Rightarrow \frac{dy}{dt}=\left(3x^2-10x+5\right)\frac{dx}{dt} \\ \text{According to the question,} \\ \Rightarrow 2\frac{dx}{dt}=\left(3x^2-10x+5\right)\frac{dx}{dt} \\ \Rightarrow 3x^2-10x+5=2 \\ \Rightarrow 3x^2-10x+3=0 \\ \Rightarrow 3x^2-9x-x+3=0 \\ \Rightarrow 3x\left(x-3\right)-1\left(x-3\right)=0 \\ \Rightarrow \left(x-3\right)=0\mathrm{or}\left(3x-1\right)=0 \\ \Rightarrow x=3\mathrm{or}x=\frac{1}{3} \end{gathered}$$