If $f^{\text{'}}\left(x\right)=\frac{1}{1+x^2}$ for all $x$ and $f\left(0\right)=0,$ then

Important Questions on Applications of Derivatives

HARD

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 7 - Applications of Derivatives>Target Exercises>Q 59

If the equation $a{x}^2+bx+c=0$ has two positive and real roots, then the equation $a{x}^2+\left(b+6a\right)x+\left(c+3b\right)=0$ has

HARD

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 7 - Applications of Derivatives>Target Exercises>Q 60

The value of $c$ in Lagrange's theorem for the function $f\left(x\right)=\left\{\begin{array}{ll}x\cos \left(\frac{1}{x}\right),& x\ne 0\\ 0,& x=0\end{array}\right.$ in the interval $\left[-1,1\right]$ is

MEDIUM

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 7 - Applications of Derivatives>Target Exercises>Q 61

Let $f$ be a function which is continuous and differentiable for all real $x$. If $f\left(2\right)=-4$ and $f^{\text{'}}\left(x\right)\ge 6$ for all $x\in \left[2,4\right]$, then

HARD

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 7 - Applications of Derivatives>Target Exercises>Q 63

If $0<x<\frac{\pi }{2}$, then

MEDIUM

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 7 - Applications of Derivatives>Target Exercises>Q 64

The range of value of $x$ for which the function $f\left(x\right)=\frac{x}{\log x}, x>0$ and $x\ne 1$, may be decreasing, is

MEDIUM

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 7 - Applications of Derivatives>Target Exercises>Q 65

$\log x-{\tan }^{-1}x$ increases in the interval

MEDIUM

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 7 - Applications of Derivatives>Target Exercises>Q 66

If $0<x<\frac{\pi }{2}$, then

MEDIUM

JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 7 - Applications of Derivatives>Target Exercises>Q 68

If $y=x^3{\left(x-2\right)}^2,$ then the values of $x$ for which $y$ increases, are