Which of the following functions $f(x)$ are differentiable at $x = 0$? (A) $|x|$ (B) $|x - 1|$ (C) $[x]$, where $[t]$ denotes the greatest integer $\leq t$ (D) $|x + 1|$ (E) $x^2$ Choose the correct answer from the options given below:

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

If $\sqrt{y+x} + \sqrt{y-x} = a$, $a > 1$, $\frac{d^2y}{dx^2}$ is equal to :

For the function $f(x)=|\cos x|$, which of the following statements are correct?

(A) $f(x)$ is continuous for all $x \in \mathbb{R}$

(B) $f(x)$ is differentiable for all $x \in \mathbb{R}$

(C) $f(x)$ is not differentiable at $x=(2n+1)\dfrac{\pi}{2}, n \in \mathbb{Z}$

(D) $f(x)$ is not differentiable at $x=n\pi, n \in \mathbb{Z}$

Choose the correct answer from the options given below:

Let $f(x) = |x-1| + |x-2|, x \in R$. Then which of the following statements is/are TRUE?

(A) $f$ is continuous everywhere.

(B) $f$ is continuous everywhere except at $x = 1$ and $x = 2$

(C) $f$ is not differentiable at $x = 1$ and $x = 2$

(D) $f$ is differentiable at $x = 1$ and $x = 2$

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 $x = t^2$ and $y = t\sqrt{t}$, then $\dfrac{d^2y}{dx^2}$ at $t = \dfrac{1}{4}$ is :