If f(x⁵) = 5x³, then f'(x) is equal to

• $\frac{3}{\sqrt[5]{{\mathrm{x}}^2}}$

• $\frac{3}{\sqrt[5]{\mathrm{x}}}$

• $\frac{3}{\mathrm{x}}$

• $\sqrt[5]{\mathrm{x}}$

f(x) = $$\begin{gathered} \left\{2\mathrm{a} - \mathrm{x} \mathrm{in} - \mathrm{a} < \mathrm{x} < \mathrm{a} \\ 3\mathrm{x} - 2\mathrm{a} \mathrm{in} \mathrm{a} \le \mathrm{x}\right\} \end{gathered}$$

Then, which of the following is true ?

• f(x) is discontinuous at x = a

• f(x) is not differentiable at x = a

• f(x) is differentiable at x  $\ge$ a

• f(x) is continuous at all x <a

If f(x) = be^ax + ae^bx, then f''(0) is equal to

• 0

• 2ab

• ab(a + b)

• ab

Th function f(x) = $\frac{\log \left(1 +\hspace{0.17em}\mathrm{ax}\right) - \log \left(1 - \mathrm{bx}\right)}{\mathrm{x}}$is not defined at x = 0. The value which should be assigned to f at x = 0 so that it is continuousat x = 0 is 

• a - b

• a + b

• log(a) + log(b)

• 0

If f(x) = 1 + nx + $\frac{\mathrm{n}\left(\mathrm{n} - 1\right)}{2}{\mathrm{x}}^2$ + $\frac{\mathrm{n}\left(\mathrm{n} - 1\right)\left(\mathrm{n} - 2\right)}{6}{\mathrm{x}}^3$ + ... + xⁿ, then f''(1) is equal to

• n(n - 1)2^{n - 1}

• (n - 1)2^{n - 1}

• n(n - 1)2^{n - 2}

• n(n - 1)2ⁿ

If f(x) = ${\log }_{{\mathrm{x}}^2}\left({\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)$, then f'(x) at x = e is

• 1

• $\frac{1}{\mathrm{e}}$

• $\frac{1}{2\mathrm{e}}$

• 0

If f(x) = $\frac{\mathrm{g}\left(\mathrm{x}\right) + \mathrm{g}\left(- \mathrm{x}\right)}{2} + \frac{2}{{\displaystyle {\left[\mathrm{h}\left(\mathrm{x}\right) + \mathrm{h}\left(- \mathrm{x}\right)\right]}^{- 1}}}$ where g and h are differentiable function, then f'(0)

• 1

• $\frac{1}{2}$

• $\frac{3}{2}$

• 0

288.

The function f(x) = [x], where [x] denotes the greatest integer not greater than x , is

• continuous for all non-integral values of x

• continuous only at positive integral values of x

• continuous for all real values of x

• continuous only at rational values of x

If the three function f(x), g(x) and h(x) are such that h(x) = f(x) g(x) and f'(x) g'(x) = c where c is constant, then

$\frac{\mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{x}\right)} + \frac{\mathrm{g}\text{'}\text{'}\left(\mathrm{x}\right)}{\mathrm{g}\left(\mathrm{x}\right)} + \frac{2\mathrm{c}}{\mathrm{f}\left(\mathrm{x}\right) . \mathrm{g}\left(\mathrm{x}\right)}$ is equal to

• h'(x) . h''(x)

• $\frac{\mathrm{h}\left(\mathrm{x}\right)}{\mathrm{h}\text{'}\text{'}\left(\mathrm{x}\right)}$

• $\frac{\mathrm{h}\text{'}\text{'}\left(\mathrm{x}\right)}{\mathrm{h}\left(\mathrm{x}\right)}$

• $\frac{\mathrm{h}\left(\mathrm{x}\right)}{\mathrm{h}\text{'}\left(\mathrm{x}\right)}$

The derivative of e^ax cos(bx) with respect x is re^ax cos(bx) ${\tan }^{-1}\left(\frac{\mathrm{b}}{\mathrm{a}}\right)$ when a>0,b>0, then a value of r, is

• $\sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2}$

• $\frac{1}{\sqrt{\mathrm{ab}}}$

• ab

• a + b