Condition for Strictly Increasing

Q. Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants):
f(x)= $(px+q)(\frac{r}{x}+s)$

Q. Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
f(x)= $(ax+b{)}^n(cx+d{)}^m$

Q. Let $f\left(x\right)=x+\ln x-x\ln xx\in (0,\infty )$

• Column 1 contains information about zeros of $f\left(x\right)$, $f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right)$.
• Column 2 contains information about the limiting behavior of $f\left(x\right)$, $f^{\prime }\left(x\right)$ and $f^{\prime \prime }\left(x\right)$ at infinity.
• Column 3 contains information about increasing/decreasing nature of $f\left(x\right)$ and $f^{\prime }\left(x\right)$.

​​​​​​$\begin{array}{ccc}Column1& Column2& Column3\\ \left(I\right)f\left(x\right)=0\text{for some}x\in (1,e^2)& \left(i\right)\underset{x\to \infty }{lim}f\left(x\right)=0& \left(P\right)f\text{is increasing in}(0,1)\\ \left(II\right)f^{\prime }\left(x\right)=0\text{for some}x\in (1,e)& \left(ii\right)\underset{x\to \infty }{lim}f\left(x\right)=-\infty & \left(Q\right)f\text{is decreasing in}(e,e^2)\\ \left(III\right)f^{\prime }\left(x\right)=0\text{for some}x\in (0,1)& \left(iii\right)\underset{x\to \infty }{lim}{f}^{\prime }\left(x\right)=-\infty & \left(R\right)f^{\prime }\text{is increasing in}(0,1)\\ \left(IV\right)f^{\prime \prime }\left(x\right)=0\text{for some}x\in (1,e)& \left(iv\right)\underset{x\to \infty }{lim}{f}^{\prime \prime }\left(x\right)=0& \left(S\right)f^{\prime }\text{is decreasing in}(e,e^2)\end{array}$

Which of the following options is the only INCORRECT combination?

1. $\left(I\right)\left(iii\right)\left(P\right)$
2. $\left(II\right)\left(iv\right)\left(Q\right)$
3. $\left(III\right)\left(i\right)\left(R\right)$
4. $\left(I\right)\left(iii\right)\left(P\right)$

Q. If $f:R\to R$ is a differentiable function such that $f^{\prime }\left(x\right)>2f\left(x\right)$ for all $x\in R$, and $f\left(0\right)=1,$ then

1. $f\left(x\right)$ is increasing in$(0,\infty )$
2. $f\left(x\right)$ is decreasing in$(0,\infty )$
3. $f\left(x\right)>e^{2x}$ in $(0,\infty )$
4. $f^{\prime }\left(x\right)<e^{2x}$ in $(0,\infty )$