31.

If f(x) is an odd differentiable function defined on $\left(- \infty , \infty \right)$ such that f'(3) = 2, then f'(- 3) is equal to

• 0

• 1

• 2

• 4

If f(x) = ${\tan }^{-1}\left(\frac{\log \left({\displaystyle \frac{\mathrm{e}}{{\mathrm{x}}^2}}\right)}{\log \left({\mathrm{ex}}^2\right)}\right) + {\tan }^{-1}\left(\frac{3 + 2\log \left(\mathrm{x}\right)}{1 - 6\log \left(\mathrm{x}\right)}\right)$

• x²

• x

• 1

• 0

The number of points at which the function $\mathrm{f}\left(\mathrm{x}\right) = \max \left\{\mathrm{a} - \mathrm{x}, \mathrm{a} + \mathrm{x}, \mathrm{b}\right\}, - \infty < \mathrm{x} < \infty , 0 < \mathrm{a} < \mathrm{b}$ cannot be differentiable, is

• 0

• 1

• 2

• 3

If f(x) is a function such that f'(x) = (x - 1)²(4 - x), then

• f(0) = 0

• f(x) is increasing in (0, 3)

• x = 4 is a critical point of f(x)

• f(x) is decreasing in (3, 5)

Let f: $\mathrm{R} \to \mathrm{R}$ be a continuous function which satisfies f(x) = ${\int }_0^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)d\mathrm{t}$. Then, the value of $\mathrm{f}\left({\log }_{\mathrm{e}}\left(5\right)\right)$ is

• 0

• 2

• 5

• 3

Let f: [2, 2] $\to$ R  be a continuous function such that f(x) assumes only irrational values. If $\mathrm{f}\left(\sqrt{2}\right) = \sqrt{2}$, then

• f(0) = 0

• $\mathrm{f}(\sqrt{2} - 1) = \sqrt{2} - 1$

• $\mathrm{f}(\sqrt{2} - 1) = \sqrt{2} + 1$

• $\mathrm{f}(\sqrt{2} - 1) = \sqrt{2}$

Let [x] denotes the greatest integer less than or equal to x. Then, the value of $\mathrm{\alpha } \mathrm{f}$ · which the function

f(x) = $$\begin{gathered} \left\{\frac{\sin \left[- {\mathrm{x}}^2\right]}{\left[- {\mathrm{x}}^2\right]}, \mathrm{x} \ne 0 \\ \mathrm{\alpha }, \mathrm{x} = 0\right\} \end{gathered}$$ is continuous at x = 0, is

• $\mathrm{\alpha } = 0$

• $\mathrm{\alpha } = \sin \left(- 1\right)$

• $\mathrm{\alpha } = \sin \left(1\right)$

• $\mathrm{\alpha } = 1$

For all real values of a₀, a₁, a₂, a₃ satisfying ${\mathrm{a}}_0 + \frac{{\mathrm{a}}_1}{2} + \frac{{\mathrm{a}}_2}{3} + \frac{{\mathrm{a}}_3}{4} = 0$, the equation a₀ + a₁x + a₂x + a₃x³ = 0 has a real root in the interval

• [0, 1]

• [- 1, 0]

• [1, 2]

• [- 2, - 1]

$$\begin{gathered} \mathrm{Let} \mathrm{f}:\hspace{0.17em}\mathrm{R} \to \mathrm{R} \mathrm{be} \mathrm{defined} \mathrm{as} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{0, \mathrm{x} \mathrm{is} \mathrm{irrational} \\ \sin \left|\mathrm{x}\right|, \mathrm{x} \mathrm{is} \mathrm{rational}\right\} \\ \mathrm{Then}, \mathrm{which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{true}? \end{gathered}$$

• f is discontinuous for all x

• f is continuous for all x

• f is discontinuous at x = k$\mathrm{\pi }$, where k is an integer

• f is continuous at x = k$\mathrm{\pi }$, where k is an integer

If $\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{axe}}^{\mathrm{x}} - \mathrm{blog}\left(1 + \mathrm{x}\right)}{{\mathrm{x}}^2} = 3$, then the value of a and b are, respectively

• 2, 2

• 1, 2

• 2, 1

• 2, 0