Embibe Experts Solutions for Exercise 4: EXERCISE 6.4

On the interval $\left[0,1\right]$, the function  $x^{25}{\left(1-x\right)}^{75}$  takes its maximum value at the point 

The two positive numbers $x$ and $y$ such that $x+y=60$ and $x{y}^3$ is maximum, is given by

Two sides of a triangle are given. The angle between them such that the area is maximum, is given by 

The rate of working of an engine is given by $R=15v+\frac{6000}{v}$, where $0<v<30$ and $v$ is the speed of engine. The value of $v$ for which $R$ is least, is given by

Which fraction exceeds its $p^{\mathrm{th}}$ power by the greatest number possible?

Maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3} \mathrm{cm}$, is given by

The largest possible area of a right angled triangle whose hypotenuse is $5 \mathrm{cm}$ is given by

The point on the curve $y^2=4x$, which is nearest to the point $\left(2, -8\right)$, is