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
$\frac{6dy}{dx}=3x^2$
$\Rightarrow \frac{dy}{dx}=\frac{1}{2}{x}^2$
$\because \Delta y=\frac{dy}{dx}\Delta x$
$\Rightarrow 8\Delta x=\frac{1}{2}{x}^2\Delta x$
$\Rightarrow x^2=16$
$\Rightarrow x=\pm 4$
When $x=4,6y=(4{)}^3+2$
$\Rightarrow 6y=66$
$\Rightarrow y=11$
Hence, required point is $(4,11)$