Question

The point on the curve $6y = x^{3} + 2$ at which $y$ - co-ordinate is changing $8$ times as fast as $x$- co-ordinate is ______________

• A. $(4, 11)$
• B. $ (4, - 11)$
• C. $ (-4, 11)$
• D. $(-4, - 11)$

Answer 1

Answer: (A)

Given, $6y=x^{3}+2\, \dots(i)$
and $ \Delta y=8 \Delta x$
On differentiating both sides of Eq. (i) w.r.t. $x$, we get
$\frac{6 dy}{dx}=3 x^{2}$
$ \Rightarrow \frac{dy}{dx}=\frac{1}{2} x^{2} $
$\because \Delta y=\frac{d y}{d x} \Delta x$
$ \Rightarrow 8 \Delta x=\frac{1}{2} x^{2} \Delta x$
$\Rightarrow x^{2}=16$
$ \Rightarrow x=\pm \,4$
When $x=4, 6y=(4)^{3}+2$
$\Rightarrow 6y=66 $
$\Rightarrow y =11 $
Hence, required point is $(4,11)$