For the function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = \lvert x-3 \rvert + \lvert x+4 \rvert$, Where $\mathbb{R}$ is set of real numbers, which of the following statements are correct?

(A) $f(x)$ is continuous as well as differentiable at $x=3$.

(B) $f(x)$ is continuous but not differentiable at $x=3$ and $x=-4$

(C) $f(x)$ is neither continuous nor differentiable at $x=3$

(D) $f(x)$ is continuous on $\mathbb{R}$

Choose the correct answer from the options given below:

For the function $f(x) = \sqrt{x}$, which of the following statements are true?

(A) $f(x)$ is continuous for $x \geq 0$.

(B) $f(x)$ is continuous for $x \in R$ (Set of real numbers).

(C) $f(x)$ is differentiable for $x > 0$.

(D) $f(x)$ is differentiable for $x \in R$ (Set of real numbers).

Choose the correct answer from the options given below:

If $x^m y^n = (x + y)^{m+n}$, then $\frac{d^2y}{dx^2}$ is equal to:

If $x = at^2$ and $y = 2at$, then the value of $\frac{d^2y}{dx^2}$ is

( where t is a parameter )

If $f(x) = \begin{cases} \frac{x^2+3x-10}{x-2}, & x \neq 2 \\ 2k+1, & x = 2 \end{cases}$ is continuous at $x = 2$, then the value of $k$ is

If $e^y(x + 1) = 1$ and $\frac{d^2y}{dx^2} = k(\frac{dy}{dx})^2$, then k is equal to

If $y = \frac{1}{x+1}$, then $\frac{d^2y}{dx^2}$ at $x = 2$ is: