The domain of ${\sin }^{-1}\left[{\log }_3\left(\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)\right]$ is

• [1, 9]

• [- 1, 9]

• [- 9, 1]

• [- 9, - 1]

The value of m for which the function f(x) = $$\begin{gathered} \left\{{\mathrm{mx}}^2, \mathrm{x} \le 1 \\ 2\mathrm{x}, \mathrm{x} > 1\right\} \end{gathered}$$, is differentiable at x = 1, is

• 0

• 1

• 2

• does not exist

If y = (1 + x^1/4)(1 + x^1/2)(1 - x^1/4), then dy/dx is equal to

• 1

• - 1

• x

• $\sqrt{\mathrm{x}}$

If y = $\log \left(\log \left(\mathrm{x}\right)\right), \mathrm{then} {\mathrm{e}}^{\mathrm{y}}\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{1}{\mathrm{xlog}\left(\mathrm{x}\right)}$

• $\frac{1}{\mathrm{x}}$

• $\frac{1}{\log \left(\mathrm{x}\right)}$

• e^y

For the function f(x) = x² - 6x + 8, $2 \le \mathrm{x} \le 4$, the value of x for which f'(x) vanishes, is

• 9/4

• 5/2

• 3

• 7/2

The function f(x) = cot^-1(x) + x, increases in the interval

• $\left(1, \infty \right)$

• $\left(- 1, \infty \right)$

• $\left(- \infty , \infty \right)$

• $\left(0, \infty \right)$

For all real values of x, increasing function is

• x^{- 1}

• x²

• x³

• x⁴

Maximum value of f(x) = sin(x) + cos(x) is

• 1

• 2

• $\frac{1}{\sqrt{2}}$

• $\sqrt{2}$

The greatest value of f(x) = (x + 1)^1/3 - (x - 1)^1/3 on [0, 1] is

• 1

• 2

• 3

• 1/3

20.

If $\mathrm{y} = {\left(\mathrm{x} + \sqrt{1 + {\mathrm{x}}^2}\right)}^{\mathrm{n}}$, then $\left(1 + {\mathrm{x}}^2\right)\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• n²y

• - n²y

• - y

• 2x²y