Question

If the rate of change in $y=\frac{x^3}{3}-2x^2+5x+7$ is two times the rate of change in $x$, then $x=$

• A. $2,3$
• B. $1,3$
• C. $1,2$
• D. $2,5$

Answer 1

Answer: (B)

$1,3$

We have,

$\begin{array}{c}y=\frac{x^3}{3}-2x^2+5x+7\\ \frac{dy}{dx}=\frac{3x^2}{3}\frac{dx}{dt}-2\frac{2dx}{dt}+5\frac{dx}{dt}\\ \frac{2x}{dt}=\frac{3x^2}{3}\frac{2dx}{dt}-4x\cdot \frac{dx}{dt}+\frac{5dx}{dt}\\ 2=\frac{3x^2}{3}-4x+5\\ \frac{3x^2}{3}-4+3=0\\ x^2-4x+3=0\\ \left(x-1\right)\left(x-3\right)=0\\ x=1,3\end{array}$

Hence, this is the answer.