The function f(x) = $\log \left(1 + \mathrm{x}\right) - \frac{2\mathrm{x}}{2 +\hspace{0.17em}\mathrm{x}}$ is increasing on

• $\left(0, \infty \right)$

• $\left(- \infty , 0\right)$

• $\left(- \infty , \infty \right)$

• None of the above

If f(x) = $\mathrm{kx} - \sin \left(\mathrm{x}\right)$, then sin(x)is monotonically increasing, then

• k > 1

• k > - 1

• k < 1

• k < - 1

163.

The maximum area of the rectangle that can be inscribed in a circle of radius r, is

• ${\mathrm{\pi r}}^2$

• r²

• $\raisebox{1ex}{${\mathrm{\pi r}}^2$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• 2r²

The velocity of a particle at time t is given by the relation v = 6t - $\frac{{\mathrm{t}}^2}{6}$. The distance traveled in 3 s is, if s = 0 at t = 0

• $\frac{39}{2}$

• $\frac{57}{2}$

• $\frac{51}{2}$

• $\frac{33}{2}$

The maximum value of function x³ - 12x² + 36x + 17 in the interval [1, 10] is

• 17

• 177

• 77

• None of these

The abscissae of the points, where the tangent is to curve y = x³ - 3x² - 9x + 5 is parallel to x-axis, are

• x = 0 and 0

• x = 1 and - 1

• x = 1 and - 3

• x = - 1 and 3

The equation of motion of a particle moving along a straight line is s = 2t³ - 9t² + 12t, where the units of s and t are centimetre and second. The acceleration of the particle will be zero after

• $\frac{3}{2}\mathrm{s}$

• $\frac{2}{3}\mathrm{s}$

• $\frac{1}{2}\mathrm{s}$

• 1 s

The equation of the tangent to the curve y = 4xe^x at $\left(- 1, \frac{- 4}{\mathrm{e}}\right)$

• y = - 1

• $\mathrm{y} = - \frac{4}{\mathrm{e}}$

• x = - 1

• $\mathrm{x} = \frac{- 4}{\mathrm{e}}$

The equation of tangent to the curve y² = ax² + b at point (2, 3) is y = 4x - 5, then the values of a and b are

• 3, - 5

• 6, - 5

• 6, 15

• 6, - 15

For all real x, the minimum value of $\frac{1 - \mathrm{x} + {\mathrm{x}}^2}{1 + \mathrm{x} + {\mathrm{x}}^2}$ is

• 0

• 1/3

• 1

• 3