A particle moves along the curve $$6y = x^3 +2$$. Find the points on the curve at which the $$y$$-coordinate is changing $8$ times as fast as the $$x$$-coordinate.

$\begin{array}{1 1} \big(-4,\frac{-31}{3}\big)\; and (4,11) \\ \big(-4,\frac{31}{3}\big)\; and\;(4,11) \\\big(-4,\frac{-31}{3}\big)\; and \;(-4,11) \\ \big(-4,\frac{-31}{3}\big)\; and \;(4,-11) \end{array}$

Toolbox:
• If $y=f(x)$,then $\large\frac{dy}{dx}$ measures the rate of change of $y$ w.r.t $x$.
• $\big(\large\frac{dy}{dx}\big)_{x=x_0}$ represents the rate of change of $y$ w.r.t $x$ at $x=x_0$
Step 1:
Let the required point be $P(x,y)$.It is given that rate of change of $y$-coordinate =8(Rate of change of $x$-coordinate)
(i.e)$\large\frac{dy}{dt}$$=8.\large\frac{dx}{dt} It is given 6y=x^3+2 On differentiating w.r.t t on both sides, 6\large\frac{dy}{dt}$$=3x^2\large\frac{dx}{dt}$$+2 \Rightarrow 6.\big(8\large\frac{dx}{dt}\big)=$$3x^2\large\frac{dx}{dt}$
$\Rightarrow 3x^2=48$
$\Rightarrow x^2=16$
$\Rightarrow x=\pm 4$
Step 2:
When $x=4$,then $6y=4^3+2$
$\Rightarrow 66$
Therefore $y=11$
When $x=-4$,then $6y=(-4)^3+2$
$\Rightarrow -62$
$y=\large\frac{-31}{3}$
Hence the required points are $\big(-4,\large\frac{-31}{3}\big)$ and $(4,11)$.