## Limits, Continuity and Differentiability Questions and Answers Part-14

1. Let f be a differentiable function on R taking non negative values such that $\lim_{x \rightarrow 0}\frac{f'\left(x\right)}{x}$   exists and satisfy
$f\left(x\right)f\left(y\right)=f\left(x_{1}\right)f\left(y_{1}\right)$
For all $x,y,x_{1},y_{1}$     such that $x^{2}+y^{2}=x_1^2+y_1^2$    such that f (0) = 1, then
a) f is infinitely many times differentiable
b) f is an exponetial function
c) $\lim_{x \rightarrow 0}f\left(x\right)=1$
d) All of the Above

Explanation: Taking logarithm of both sides,

2. Let [x] denote the greatest integer less than or equal to x. If $f\left(x\right)=\left[x\sin\pi x\right]$    , then f (x) is
a) continuous at x = 0
b) continuous in (–1, 0)
c) differentiable in (– 1, 1)
d) All of the Above

Explanation: By definition of [x], we have f(x) = [x sin $\pi$ x]

3. The function $f\left(x\right)=\begin{cases}\mid x-3 \mid& x \geq 1\\\frac{x^{2}}{4}-\frac{3x}{2}+\frac{13}{4} & x < 1\end{cases}$
is
a) continuous at x = 1
b) continuous at x = 3
c) differentiable at x = 1
d) All of the Above

Explanation: The given function can be written as

4. If $f\left(x\right)=\begin{bmatrix}x^{n} &\sin x & -\cos x \\n! &\sin\left(n\pi/2\right)& \cos\left(n\pi/2\right)\\a &a^{2} & a^{3} \end{bmatrix}$
the value of $\frac{d^{n}}{dx^{n}}\left(f\left(x\right)\right)$    at x = 0 for n = 2m + 1 is
a) independent of a
b) 0
c) $a^{6}$
d) Both a and b

Explanation:

5. If $f\left(x\right)=\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}$
then
a) f is a differentiable at all points of its domain except x= 4
b) f is differentiable on $\left(2,\infty\right)\sim\left\{4\right\}$
c) f' (x) = 0 for all $x\epsilon$ [2,4)
d) All of the Above

Explanation:

6. If f'(5) = 4 then $\lim_{x \rightarrow \infty}\frac{f\left(5+h^{2}\right)-f\left(5-h^{2}\right)}{2h^{2}}$
is
a) 2
b) 4
c) 3
d) 1/4

Explanation:

7. If $y=\tan^{-1}\left(\frac{\log ex^{-3}}{\log ex^{3}}\right)+\tan^{-1}\left(\frac{3+3\log x}{1-9\log x}\right)$
then $\frac{d^{2} y}{dx^{2}}$   is
a) 2
b) 1
c) 0
d) -1

Explanation:

8. If $f\left(x\right)=\frac{\tan x+\sec x -1}{\tan x-\sec x+1}$     then $f'\left(\pi/4\right)$   is equal to
a) $\sqrt{2}\left(\sqrt{2}-1\right)$
b) $\sqrt{2}+1$
c) $\sqrt{2}\left(\sqrt{2}+1\right)$
d) $\sqrt{2}-1$

Explanation:

9. If $f\left(x\right)=x^{1/x}$    then f'' (e) is equal to
a) $e^{1/\left({e-3}\right)}$
b) $e^{1/e}$
c) $e^{1/\left({e-2}\right)}$
d) $e^{1/e-3}$

10. If $y=f\left(\frac{3x+4}{5x+6}\right)$    and $f"\left(x\right)=\tan x^{2}$    then $\frac{dy}{dx}$  is equal to
a) $-2\tan\left(\frac{3x+4}{5x+6}\right)^{2}\times\frac{1}{\left(5x+6\right)^{2}}$
b) $f\left(\frac{3\tan x^{2}+3}{5\tan x^{2}+6}\right) \tan x^{2}$
c) $\tan x^{2}$
d) $\tan \left(\frac{3x+4}{5x+6}\right)^{2}$