Applications of Derivatives Questions with Solutions

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 equation of the tangent line to the curve $y=2xsinx$ at the point $\left(\frac{\pi }{2},\pi \right)$ is

What is the largest area of an isosceles triangle with two edges of length 3?

The critical point and nature for the function $f\left(x,y\right)=x^2-2x+2y^2+4y-2$ is

If the radius of a circle changes at the rate of $-\frac{2}{\pi }$ m/s, at what rate does the area of the circle change when the radius is 10 m?

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 $f\left(x\right)=x^{2/3}{\left(6-x\right)}^{1/3}$. Which of the following statements is false?

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