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

Consider the following statements for the curve $f(x) = x|x|$ and find that which of the following(s) are/is correct :

(A) $f(x) = x|x|$ is differentiable at $x = 0$.

(B) $f(x) = x|x|$ is continuous at $x = 0$ but not differentiable at $x = 0$.

(C) $f(x) = x|x|$ has point of infection at $x = 0$.

(D) $f(x) = x|x|$ is symmetrical about y-axis.

Choose the correct answer from the options given below :

$f(x) = [x]$, where $[\,]$ represents greatest integer function.

(A) For $2 \leq x < 3$, $[x] = 3$

(B) For $2 \leq x < 3$, $[x] = 2$

(C) Right hand derivative of f at $x = 2$ is not defined

(D) Left hand derivative of f at $x = 2$ is zero

(E) $f(x)$ is not differentiable at $x = 2$

Choose the correct answer from the options given below :

The function $f(x) = |x - 1|$ is

If $y = (\log_e x)^{\log_e x}$, then $\dfrac{dy}{dx}$ is equal to:

If $x = 2t^2$ and $y = 3t^3$, then the value of $\frac{d^2y}{dx^2}$ at $t = 3$ is :

If $f(x) = \begin{cases} \dfrac{1}{|x|}, & x \geq 1 \\ ax^2+b, & x < 1 \end{cases}$ is differentiable at $x=1$ then