If $y=\frac{\log x}{x}$, then $\frac{d^2y}{dx^2}$ is equal to(Consider $\log_e x = \log x$):

If $f(x) = \begin{cases} \frac{\tan(\frac{\pi}{4} - x)}{\cot 2x} & , x ≠ \frac{\pi}{4} \\ 2K + 1 & , x = \frac{\pi}{4} \end{cases}$ is continuous at $x = \frac{\pi}{4}$, then the value of K is equal to

For the function $f(x) = \min\{|x|, |x-2|\}$ which of the following are TRUE?

A. Function $f(x)$ is continous on $\mathbb{R}$

B. Function $f(x)$ is differentible on $\mathbb{R}$

C. Function $f(x)$ is differentible on $\mathbb{R} \sim \{0,1,2\}$

D. Function is continous on $\mathbb{R}$ but not differentiable at $x = \dfrac{1}{2}$

Choose the correct answer from the options given below:

If $f(x) = \begin{cases} ax - 1 & {if } x \ > 1\\ \ 2x + 1 & {if } x < 1 \end{cases}$ is continuous at $x = 1$, then $a$ equals

If the function $f(x) = \dfrac{(4^x-1)^3}{\sin\dfrac{x}{4}\cdot\log\left(1+\dfrac{x^2}{3}\right)}$ is continuous at $x = 0$, then the value of $f(0)$ is: (Take $\log_e x = \log x$)

If $y = 500 e^{7x} + 600 e^{-7x}$ and $\frac{d^2 y}{dx^2} = ky$, then the value of k is :

Differentiation of $\log[\log(\log x^5)]$ with respect to $x$ is