$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\left[\mathrm{x}\right] + \left[- \mathrm{x}\right], \mathrm{when} \mathrm{x} \ne 2 \\ \mathrm{\lambda }, \mathrm{when} \mathrm{x} = 2\right\} \end{gathered}$$

If f (x) is continuous at x = 2 then, the value of $\mathrm{\lambda }$ will be

• - 1

• 1

• 0

• 2

For what values of x, the function f(x) = x⁴ - 4x³ + 4x² + 40 is monotonic decreasing ?

• 0 < x < 1

• 1 < x < 2

• 2 < x < 3

• 4 < x < 5

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

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

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

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

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

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

f(x) = x + $\left|\mathrm{x}\right|$ is continuous for

• $\mathrm{x} \in \left(- \infty , \infty \right)$

• $\mathrm{x} \in \left(- \infty , \infty \right) - \left\{0\right\}$

• only x > 0

• no value of x

686.

The second order derivative of $\mathrm{a} {\sin }^3\left(\mathrm{t}\right)$ with respect to ${\cos }^3\left(\mathrm{t}\right)$ at t = $\frac{\mathrm{\pi }}{4}$ is

• 2

• $\frac{1}{12\mathrm{a}}$

• $\frac{4\sqrt{2}}{3\mathrm{a}}$

• $\frac{3\mathrm{a}}{4\sqrt{2}}$

If x² + y² = 1, then

• yy'' - (2y')² + 1 = 0

• yy'' + (y')² + 1 = 0

• yy'' - (y')² - 1 = 0

• yy'' + 2(y')² + 1 = 0

The Rolle's theorem is applicable in the interval $- 1 \le \mathrm{x} \le 1$ for the function

• f(x) = x

• f(x) = x²

• f(x) = 2x³ + 3

• f(x) = $\left|\mathrm{x}\right|$

If $\mathrm{x} = \sin \left(\mathrm{t}\right), \mathrm{y} = \sin \left(2\mathrm{t}\right)$, prove that

$\left(1 - {\mathrm{x}}^2\right)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} - \mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}} + 4\mathrm{y}$ = 0

If f is differentiable at x = a, find the value of

$\underset{\mathrm{x}\to \mathrm{a}}{\lim }\frac{{\mathrm{x}}^2\mathrm{f}\left(\mathrm{a}\right) - {\mathrm{a}}^2\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{x} - \mathrm{a}}$