Applications of Derivatives Questions with Solutions

Find the acute angle at which the curves $y={\left(x-2\right)}^2$ and $y=-4+6x-x^2$ intersect.

The population of a town grows at $10$ per year. How long will it take for the population to triple?

The minimum value of $9^x+9^{1-x}$, $x\in R$ is

The local maximum value of $f\left(x\right)={\left(x-1\right)}^2{\left(x+1\right)}^3$ is equal to $\frac{2^p{3}^q}{3125}$. Then the ordered pair $\left(p,q\right)$ is

The minimum value of $px+qy$, when $xy=r^2$ and $p,q,x,y$ are positive numbers, is

If $y=alog\left|x\right|+b{x}^2+x$ has its extremum values at $x=-1$ and $x=2$, then

The rate of increase of the length of the shadow of a man $2$ m high, due to a lamp at a height of $10$ m, when he is moving away from it at $2$ m/sec, is

Water runs into a conical tank of radius $5$ ft and height $10$ ft at a constant rate of $2$ cubic ft per minute. How fast is the water level rising when the water is $6$ ft deep?

A box open at the top is made by cutting equal squares from the four corners of a $6\times 6$ m sheet and folding up the sides. The height of the box for maximum volume is:

The function $f\left(x\right)=2sinx+sin2x,x\in \left[0,2\pi \right]$ has absolute maximum and minimum at

If $f\left(x\right)=\{x^2,x\le 02sinx,x>0$, then $x=0$ is

The function $f\left(x\right)=\frac{x}{1+xtanx}$, $0\le x\le \frac{\pi }{2}$, is maximum when

Suppose that the temperature at a point $\left(x,y\right)$ on a metal plate is $T\left(x,y\right)=4x^2-4xy+y^2$. An ant, walking on the plate, traverses a circle of radius 5 centred at the origin. What is the highest temperature encountered by the ant?

A condition that $x^3+a{x}^2+bx+c$ may have no extremum is

A function $f:\left(0,\pi \right)\to R$ defined by $f\left(x\right)=2sinx+cos2x$ has

The tangent at the point $\left(2,-2\right)$ to the curve $x^2{y}^2-2x=4\left(1-y\right)$ does not pass through the point

Consider the function $f\left(x\right)=-x^3+3x^2+1$ if $x\le 2$; $f\left(x\right)=cosx$ if $2<x\le 4$; $f\left(x\right)=e^{-x}$ if $x>4$. Which of the following statements about $f\left(x\right)$ is true?

If the line $a^{2}x+ay+1=0$, for some real number $a$, is normal to the curve $xy=1$, then

The slope of the normal line to the curve $x=t^2+3t-8$, $y=2t^2-2t-5$ at the point $\left(2,-1\right)$ is