Embibe Experts Solutions for Chapter: Applications of Derivatives, Exercise 1: ICSE-2020

The average cost function associated with producing and marketing $x$ units of an item is given by $AC=2x-11+\frac{50}{x}$. Find the range of values of the output $x$, for which $AC$ is increasing.

Prove that the function $f\left(x\right)=x^3-6x^2+12x+5$ is increasing on $R$.

A $13 \mathrm{m}$ long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of $2 \mathrm{m}/\mathrm{s}$. How fast is the height on the wall decreasing when the foot of the ladder is $5 \mathrm{m}$ away from the wall?

The volume of a closed rectangular metal box with a square base is $4096 {\mathrm{cm}}^3$. The cost of polishing the outer surface of the box is $4 \mathrm{rupees} \mathrm{per} {\mathrm{cm}}^2$. Find the dimensions of the box for the minimum cost of polishing it.

The edge of a variable cube is increasing at the rate of $10 \mathrm{cm}/\sec$. How fast is the volume of the cube increasing when the edge is $5 \mathrm{cm}$ long?

The equation of tangent at $\left(2,3\right)$ on the curve $y^2=p{x}^3+q$ is $y=4x-7$. Find the values of $p$ and $q$.

Show that the radius of a closed right circular cylinder of a given surface area and maximum volume is equal to half of its height.

Prove that the area of right-angled triangle of given hypotenuse is maximum when the triangle is isosceles.