Amit M Agarwal Solutions for Chapter: Monotonicity, Maxima and Minima, Exercise 4: Proficiency in 'Monotonicity, Maxima and Minima' Exercise 1

Let $f\left(x\right)=ab\mathrm{s}\mathrm{i}\mathrm{n}x+b\sqrt{1-a^2}\mathrm{c}\mathrm{o}\mathrm{s}x+c$, where $\left|a\right|<1, b>0,$ then

If $f\left(x\right)={\int }_{x^m}^{x^n}\frac{dt}{\mathrm{l}\mathrm{n}t},x>0$ and $n>m$, then:

$f\left(x\right)=\sqrt{x-1}+\sqrt{2-x}$ and $g\left(x\right)=x^2+bx+c$ are two given functions such that $f\left(x\right)$ and $g\left(x\right)$ attain their maximum and minimum values respectively for same value of $x$, then

If $f\left(x\right)=a^{\left\{a^{\left|x\right|}\mathrm{s}\mathrm{g}\mathrm{n}x\right\}}; g\left(x\right)=a^{\left[a^{\left|x\right|}·\mathrm{sgn}x\right]}$ for $a>0, a\ne 1$ and $x\in R,$ where $\{.\}$ and $\left[.\right]$ denotes the fractional part and integral part functions respectively, then which of the following statements hold good for the function $h\left(x\right)$, where $\left(\ln a\right)h\left(x\right)=\left(\mathrm{l}\mathrm{n}f\left(x\right)+\mathrm{l}\mathrm{n}g\left(x\right)\right)$.

For the function $f\left(x\right)=\ln \left(1-\ln x\right)$, which of the following do not hold good?

The function $f\left(x\right)=\left\{\begin{array}{ccc}x+2& ,& x<-1\\ x^2& ,& -1\le x<1\\ (x-2{)}^2& ,& x\ge 1\end{array}\right.$

A function $f$ is defined by $f\left(x\right)={\int }_0^{\pi }\mathrm{c}\mathrm{o}\mathrm{s}t \mathrm{c}\mathrm{o}\mathrm{s}\left(x-t\right)dt, 0\le x\le 2\mathrm{\pi }$, then which of the following hold(s) good

Let $f\left(x\right)=\frac{x-1}{x^2}$, then which of the following is/are correct?