Embibe Experts Solutions for Chapter: Application of Derivatives, Exercise 1: Meghalaya Board-2018

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3} \mathrm{cm}$ is $\left(500\pi \right){\mathrm{cm}}^3$.

A window is in the form of a rectangle, surmounted by a semicircular opening. The total perimeter of the window is $10$ metres. Find the dimensions of the window to admit maximum light through it.

Find the minimum value of $f\left(x\right)=(2x-1{)}^2+3$.

Show that the function $f\left(x\right)=e^{2x}$ is increasing on $R$.

Find the rate of change of the area of a circle with respect to its radius $r,$ when $r=4$ cm.

Find the slope of the tangent to the curve $y=3x^4-4x$ at  $x = 4.$

Find the equation of all lines having slope $2$ and being tangent to the curve $y+\frac{2}{x-3}=0$.

Show that $y=\log \left(1+x\right)-\frac{2x}{2+x},x>-1$, is an increasing function of $x$ throughout its domain.