If the function $f(x)=\begin{cases}3x-6, & x \le 5 \\ 4k, & x>5\end{cases}$ is continuous at $x=5$, then the value of $k$ is:

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

Differentiation of $\frac{x^3}{1 - x^3}$ with respect to $x^3$ is equal to:

If $f(x) = \begin{cases} \frac{x^2 - 9}{x - 3}, & x \neq 3 \\ 5, & x = 3 \end{cases}$ then $f(x)$ :

For the function $f(x) = |x + 1| + |x - 1| + |x - 2|$ which of the following are TRUE?

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

B. $f(x)$ is not differentiable at $x = 1$

C. $f(x)$ is differentiable on $\mathbb{R}$

D. $f(x)$ has no local maxima

Choose the correct answer from the options given below:

if $x=\sqrt{a^{\tan^{-1}t}}$ and $y=\sqrt{a^{\cot^{-1}t}}$ then which one of the following is true?

If $y=\sqrt[5]{\log_e(5x^2)}$, then $\dfrac{dy}{dx}=$

(Take $\log_e x=\log x$)