The point in the interval [0, 2], where f(x) = e^x sin(x) has maximum slope, is

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• $\frac{3\mathrm{\pi }}{2}$

The minimum value of f(x) = ${\mathrm{e}}^{\left({\mathrm{x}}^4 - {\mathrm{x}}^3 + {\mathrm{x}}^2\right)}$

• e

• - e

• 1

• - 1

In which of the following functions, Rolle's theorem is applicable

• f(x) = $\mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x}\right| \mathrm{in} - 2 \le \hspace{0.17em}\mathrm{x} \le 2$

• $\mathrm{f}\left(\mathrm{x}\right) = \tan \left(\mathrm{x}\right) \mathrm{in} 0 \le \mathrm{x} \le \mathrm{\pi }$

• $\mathrm{f}\left(\mathrm{x}\right) = 1 + {\left(\mathrm{x} - 2\right)}^{\frac{2}{3}} \mathrm{in} 1 \le \mathrm{x} \le 3$

• $\mathrm{f}\left(\mathrm{x}\right) = \mathrm{x}{\left(\mathrm{x} - 2\right)}^2 \mathrm{in} 0 \le \mathrm{x} \le 2$

54.

If y = (1 + x)(1 + x²)(1 + x⁴)...(1 + x²ⁿ) then the value of ${\left(\frac{}{}\right)}_{$ {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{\mathrm{x} = 0}$}$ is

• 0

• - 1

• 1

• 2

The function f(x) = $\sec \left[\log \left(\mathrm{x} + \sqrt{1 + {\mathrm{x}}^{{}^2}}\right)\right]$

• odd

• even

• neither odd nor even

• constant

The domain of the function f(x) = $\sqrt{{\cos }^{-1}\left(\frac{1 - \left|\mathrm{x}\right|}{2}\right)}$ is

• (- 3, 3)

• [- 3, 3]

• $\left(- \infty , - 3\right) \cup \left(3, \infty \right)$

• $(- \infty , - 3] \cup [3, \infty )$

If the line ax + by + c = 0 is a tangent to the curve xy = 4, then 

• a < 0, b > 0

• $\mathrm{a} \le 0, \mathrm{b} > 0$

• a < 0, b < 0

• $\mathrm{a} \le 0, \mathrm{b} < 0$

If the normal to the curve y = f(x) at the point (3, 4) make an angle 3$\mathrm{\pi }$/4 with the positive x-axis, then f'(3) is

• 1

• - 1

• - $\frac{3}{4}$

• $\frac{3}{4}$

The equation of normal of x² + y² - 2x + 4y - 5 = 0 at (2, 1) is

• y = 3x - 5

• 2y = 3x - 4

• y = 3x + 4

• y = x + 1

The area of the region bounded by y² = x and $\mathrm{y} = \left|\mathrm{x}\right|$ is

• $\frac{1}{3}\mathrm{sq}. \mathrm{unit}$

• $\frac{1}{6}$ sq. units

• $\frac{2}{3}$ sq. units

• 1 sq. units