Embibe Experts Solutions for Chapter: Application of Derivatives, Exercise 2: Exercise-2

Let $f\left(x\right)=\left\{\begin{array}{cc}-x^2,& x<0\\ x^2+8,& x\ge 0\end{array}\right.$ Equation of tangent line touching both branches of $y=f\left(x\right)$ is

Minimum distance between the curves $f\left(x\right)=e^x\&g\left(x\right)=\ln x$ is

The point $\left(\mathrm{s}\right)$ on the parabola $y^2=4x$ which are closest to the circle, $x^2+y^2-24y+128=0$ is/are:

If $f\left(x\right)={\sin }^{3}x+\lambda {\sin }^{2}x;-\frac{\mathrm{\pi }}{2}<x<\frac{\mathrm{\pi }}{2}$, then the interval in which $\lambda$ should lie in order that $f\left(x\right)$ has exactly one minima and one maxima

Let $f\left(x\right)=\left\{\begin{array}{cc}{x}^3-x^2+10x-5,& x\le 1\\ -2x+{\log }_2\left(b^2-2\right),& x\ge 1\end{array}\right.$ the set of values of $b$ for which $f\left(x\right)$ has greatest value at $x=1$ is given by:

For the curve $x=t^2+3t-8,$ $y=2t^2-2t-5$, at point $\left(2, -1\right)$

If $f\left(x\right)=\frac{x^2}{2-2\cos x};g\left(x\right)=\frac{x^2}{6x-6\sin x}$ where $0<x<1$, then

If $f\left(x\right)=a\ln \left|x\right|+b{x}^2+x$ has its extremum values at $x=-1$ and $x=2,$ then