Limits, Continuity and Differentiability

Which of the following is true?

Let the function \(g:\left( { - \infty ,\infty } \right) \to \left( { - {\pi \over 2},{\pi \over 2}} \right)\) be given by
\(g\left( u \right) = 2{\tan ^{ - 1}}\left( {{e^u}} \right) - {\pi \over 2}.\) Then, \(g\) is

Let \(f(x)\) be a non-constant twice differentiable function defined on \(\left( { - \infty ,\infty } \right)\)
such that \(f\left( x \right) = f\left( {1 - x} \right)\) and \(f'\left( {{1 \over 4}} \right) = 0.\) Then,

Let \(g(x) = {{{{(x - 1)}^n}} \over {\log {{\cos }^m}(x - 1)}};0 < x < 2,m\) and \(n\) are integers, \(m \ne 0,n > 0\), and let \(p\) be the left hand derivative of \(|x - 1|\) at \(x = 1\). If $$\mathop {\lim }\limits_{x \to {1^ + }} g(x) ...

Let \(f(x) = {{{x^2} - 6x + 5} \over {{x^2} - 5x + 6}}\).
Match the conditions/expressions in Column I with statements in Column II.

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For \(k > 0\), the set of all values of \(k\) for which \(k e^{x}-x=0\) has two distinct roots is

The positive value of \(k\) for which \(k e^{x}-x=0\) has only one root is

The line \(y=x\) meets \(y=k e^{\mathrm{x}}\) for \(k \leq 0\) at

Let \(f(x)=2+\cos x\) for all real \(x\).
STATEMENT - 1 : For each real \(t\), there exists a point \(c\) in \([t, t+\pi]\) such that \(f^{\prime}(C)=0\).
STATEMENT - 2 : \(f(t)=f(t+2 \pi)\) for each real \(t\).

In the following [x] denotes the greatest integer less than or equal to x.
Match the functions in Column I with the properties Column II.

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For $x>0, \mathop {\lim }\limits_{x \to 0}\left((\sin x)^{1 / x}+(1 / x)^{\sin x}\right)$ is :

If $f(x)=\min \left\{1, x^2, x^3\right\}$, then

If \(f(x-y)=f(x) \circ g(y)-f(y) \circ g(x)\) And \(g(x-y) =g(x) \circ g(y)+f(x) \circ f(y)\) for all \(x, y \in \mathrm{R}\). If right-hand derivative at \(x=0\) exists for \(f(x)\), find the derivative of \(g(x)\) at \(x=0\)