If $e^y(x+1)=1$, then

If $x = at^2, y = 2at$; then $\frac{d^2y}{dx^2}$ is equal to

Let $x = t^2, y = t^3$. Then $\frac{d^2y}{dx^2}$ is equal to

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:

Let $y = \log(x + \sqrt{x^2+1})$, and $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} = c$. Then identify the correct statements about the values of $a$, $b$ and $c$ :

(A) $a = 1 + x^2$

(B) $b = 0$

(C) $c = 0$

(D) $b = x$

(E) $c = 2$

Choose the correct answer from the options given below :

The value of k, for which the function $f(x) = \begin{cases} \frac{\sin kx}{x} + 3\cos x, & x \neq 0 \\ 7, & x = 0 \end{cases}$ is continuous at $x = 0$, is

If $f(x) = \begin{cases} \frac{k\cos x}{\pi - 2x}, & x \neq \frac{\pi}{2} \\ 3, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$, then k is :