On the curve x3=12y, the values of x for which the abscissa changes at a faster rate than the ordinate is

On the curve $x^{3} = 12 y$ , the values of $x$ for which the abscissa changes at a faster rate than the ordinate is

A
$(- 2, 2) ~ {0}$
B
$(- 3, 3) ~ {0}$
C
(1, 4)
D
(2, 4)

Solution:

$3 x^{2} \frac{d x}{d y} = 12 \Rightarrow \frac{d x}{d y} = \frac{4}{x^{2}}$ . We need $|\frac{d x}{d y}| > 1$

$\Rightarrow \frac{4}{x^{2}} > 1 \Rightarrow x^{2} < 4, x \neq 0 \Rightarrow x \in (- 2, 2) ~ {0}$

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