Match the LIST-I with LIST-II

	LIST-I Curve $y = f(x)$, where $f(x)$ is		LIST-II Nature / derivative / value of $y = f(x)$
A.	$x\vert x \vert$	I.	$y'$ is 2 for all $x \in R^+$ (set of positive real numbers)
B.	$x + \vert x \vert$	II.	$y'$ is 2 for all $x \in R^-$ (set of negative real numbers)
C.	$x - \vert x \vert$	III.	$f(-5) = -1$
D.	$\dfrac{x}{\vert x \vert}$	IV.	$(0, 0)$ is the point of inflection

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If $y = \log_e\left(\frac{e^2}{x^2}\right)$ for $x \neq 0$, then $\frac{d^2y}{dx^2}$ equals

Match the LIST-I with LIST-II

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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

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If the function $f(x)=\begin{cases} x^2, & x\le c \\ ax+b, & x>c \end{cases}$ is differentiable at $x=c$, then the relation between $b$ and $c$ is:

If $y = \left[\dfrac{2}{1+x^{b-a}+x^{c-a}} + \dfrac{2}{1+x^{c-b}+x^{a-b}} + \dfrac{2}{1+x^{a-c}+x^{b-c}}\right]^x$, then $\dfrac{dy}{dx}$ is equal to :

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 $x = 2$

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