The maximum value of $\mathrm{f}\left(\mathrm{x}\right) = \frac{\log \left(\mathrm{x}\right)}{\mathrm{x}}\left(\mathrm{x} \ne 0, \mathrm{x} \ne 1\right)$ is

• e

• $\frac{1}{\mathrm{e}}$

• e²

• $\frac{1}{{\mathrm{e}}^2}$

If the volume of spherical ball is increasing at the rate of 4$\mathrm{\pi }$ cm³/s, then the rate of change of its surface area when the volume is 288 $\mathrm{\pi }$ cm³, is

• $\frac{4}{3}\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $\frac{2}{3}\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $4\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $2\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

The equation of displacement of a particle is x(t) = 5t² - 7t + 3. The acceleration at the moment when its velocity becomes 5 m/sec is

• 3 m/sec²

• 7 m/sec²

• 10 m/sec²

• 8 m/sec²

The mean value of the function $\mathrm{f}\left(\mathrm{x}\right) = \frac{2}{{\mathrm{e}}^{\mathrm{x}} + 1}$ on the interval [0, 2] is

• $2 - {\log }_{\mathrm{e}}\left(\frac{2}{{\mathrm{e}}^2 + 1}\right)$

• $2 + {\log }_{\mathrm{e}}\left(\frac{2}{{\mathrm{e}}^2 + 1}\right)$

• $2 + {\log }_{\mathrm{e}}\left(\frac{2}{{\mathrm{e}}^2 - 1}\right)$

• $- 2 + {\log }_{\mathrm{e}}\left(\frac{2}{{\mathrm{e}}^2 - 1}\right)$

The function $\mathrm{y} = \sqrt{2\mathrm{x} - {\mathrm{x}}^2}$

• increases in (0, 1) but decreases in (1, 2)

• decreases in (0, 2)

• increases m (1, 2) but decreases in (0, 1)

• increases in (0, 2)

The interval in which the function $\mathrm{y} = \mathrm{x} - 2\sin \left(\mathrm{x}\right)$; $0 \le \mathrm{x} \le 2\mathrm{\pi }$ increases throughout is

• $\left(\frac{5\mathrm{\pi }}{3}, 2\mathrm{\pi }\right)$

• $\left(0, \frac{\mathrm{\pi }}{3}\right)$

• $\left(\frac{\mathrm{\pi }}{3}, \frac{5\mathrm{\pi }}{3}\right)$

• $\left(0, \frac{\mathrm{\pi }}{4}\right)$

The points of the curve y = x³ + x - 2 at which its tangent are parallel to the straight line y = 4x - 1 are

• (2, 7), (- 2, - 11)

• (0, 2), (2^1/3, 2^1/3)

• (- 2^1/3, - 2^1/3), (0, - 4)

• (1, 0), (- 1, - 4)

The equation of the normal to the curve y = - $\sqrt{\mathrm{x}}$ + 2 at the point of its intersection with the bisector of the first quadrant is

• 4x - y + 16 = 0

• 4x - y = 16

• 2x - y - 1 = 0

• 2x - y + 1 = 0

The angle at which the curve y = x² and the curve x = $\frac{5}{3}\cos \left(\mathrm{t}\right)$, y = $\frac{5}{4}\sin \left(\mathrm{t}\right)$ intersect is

• ${\tan }^{-1}\left(\frac{2}{41}\right)$

• ${\tan }^{-1}\left(\frac{41}{2}\right)$

• $- {\tan }^{-1}\left(\frac{2}{41}\right)$

• $2{\tan }^{-1}\left(\frac{41}{2}\right)$

The maximum value of the function $\mathrm{y} = 2\tan \left(\mathrm{x}\right) - {\tan }^2\left(\mathrm{x}\right)$ over $\left[0, \frac{\mathrm{\pi }}{2}\right]$ is

• $\infty$

• 1

• 3

• 2