Match **List-I** with **List-II** | List-I | List-II | | :--- | :--- | | **Function** | **Property** | | (A) $f(x) = \begin{cases} \frac{x}{\vert x \vert} & : x \neq 0 \\ 0 & : x = 0 \end{cases}$ | (I) continuous but not differentiable at $x= 0$ | | (B) $f(x) = \vert x \vert$ | (II) continuous but not differentiable at $x=1$ | | (C) $f(x) = \vert x^2 - 1 \vert$ | (III) discontinuous at $x = 0$ | | (D) $f(x) = \vert x - 1 \vert$ | (IV) continuous but not differentiable at $x = 1, -1$ | Choose the **correct** answer from the options given below:

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$\frac{d}{dx}\left(e^{2\log_e x^3}\right)$ equals

Let a function $f(x)=\alpha+(\beta^2+5\beta+6)|x|+\gamma|x|^4$, where $\alpha, \beta$ and $\gamma$ are real number. Then the function $f(x)$ is differentiable at $x=0$ if

Let $f(x) = |3-x| + (3+x)$ be a function, then which of the following statements are TRUE?

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

B. $f(x)$ is continuous at $x=-3$

C. $f(x)$ is differentiable at $x=3$

D. $f(x)$ is differentiable at $x=-3$

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The equation of the tangent to the curve $\mathrm{x}^{\frac{5}{2}}+\mathrm{y}^{\frac{5}{2}}=33$ at the point $(1,4)$ is :

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