A particle moves along the curve

Question:

A particle moves along the curve $y=x^{3}$. Find the points on the curve at which the $y$-coordinate changes three times more rapidly than the $x$-coordinate.

Solution:

According to the question,

$\frac{d y}{d t}=3 \frac{d x}{d t}$

Now,

$y=x^{3}$

$\Rightarrow \frac{d y}{d t}=3 x^{2} \frac{d x}{d t}$

$\Rightarrow 3 \frac{d x}{d t}=3 x^{2} \frac{d x}{d t}$

$\Rightarrow x^{2}=1$

$\Rightarrow x=\pm 1$

Substituting $x=\pm 1$ in $y=x^{3}$, we get

$y=\pm 1$

So the points are $(1,1)$ and $(-1,-1)$.

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