$\text{ A point is moving on }y=4-2x^2\text{. The }x\text{ coordinate of the point is decreasing }$$\mathrm{at} \mathrm{the} \mathrm{rate} \mathrm{of} 5 \text{ units per second. Find the rate at which the y-coordinate }$$\mathrm{of} \mathrm{the} p\mathrm{oint} \mathrm{is} \mathrm{changing} \mathrm{when} \text{ point is at }(1,2)\text{. }$

 

$\text{ Let }P\text{ be the moving on the curve }y=4-2x^2$

$\text{ Let }(x,y)\text{ the coordinates of }P\text{ at any point }t\text{. }$

$\begin{array}{l}\Rightarrow \frac{dx}{dt}=-5\text{ units }/\sec \\ \text{ When }(x,y)=(1,2)\text{ we have to find }\frac{dy}{dt}\text{. Diff (1) w.r.t. }\frac{dy}{dt}=-4x\frac{dx}{dt}\\ \therefore \text{ When }(x,y)=(1,2)\cdot \frac{dx}{dt}=-5,\frac{dy}{dt}=-4\left(1\right)(-5)=20\text{ units }/\sec \text{. }\end{array}$

The rate change in y is 20 units/sec. (i.e., y is increasing at the rate of 20 units/sec) 
 Hence option (D) is correct.