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.

Let $f:R\to R$ be defined by $f\left(x\right)=\left|x+1\right|e^{-x^2}$. Then which of the following statements is true?

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

The sum of two numbers is $10$. Their product will be maximum when they are

The maximum value of $\frac{logx}{x}$ is

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

Between any two real roots of the equation $e^{x}sinx=1$, the equation $e^{x}cosx=-1$ has

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 equation of the tangent at any point of the curve $x=acos2t$, $y=2\sqrt{2}asint$ with $m$ as its slope is

Find the point at which the tangent to the curve $y=\sqrt{4x-3}-1$ has its slope $\frac{2}{3}$.

The minimum value of the function $y=2x^3+36x-20$ is

The normal to the curve $y=x^3-3x+2$ at the point $\left(2,4\right)$ is

The point on the curve $y=6x-x^2$ where the tangent is parallel to the x-axis 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 angle of intersection of the cardioids $r=a\left(1+cos\theta \right)$ and $r=a\left(1-cos\theta \right)$ is

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