$$A$$ rod $$A B, 13$$ feet long moves with its ends $$A$$ and $$B$$ on two perpendicular lines $$O X$$ and $$O Y$$ respectively. When $$A$$ is 5 feet from $$O$$, it is moving away at the rate of $$3 \mathrm{feet} / \mathrm{sec}$$. At this instant, $$\mathrm{B}$$ is moving at the rate

The equation of the tangent to the curve $$y=\sqrt{9-2 x^2}$$, at the point where the ordinate and abscissa are equal, is

At present a firm is manufacturing 1000 items. It is estimated that the rate of change of production $$\mathrm{P}$$ w.r.t. additional number of worker $$x$$ is given by $$\frac{\mathrm{dp}}{\mathrm{d} x}=100-12 \sqrt{x}$$. If the firm employees 9 more workers, then the new level of production of items is

Value of $$c$$ satisfying the conditions and conclusions of Rolle's theorem for the function $$\mathrm{f}(x)=x \sqrt{x+6}, x \in[-6,0]$$ is