## Limits, Continuity and Differentiability Questions and Answers Part-11

1. Let $f\left(x\right)=\begin{cases}x^{3} \mid \cos\frac{\pi}{2x}\mid& x \neq 0\\+x^{4}\mid\sin\frac{\pi}{x}\mid\\0, & x= 0\end{cases}$
then f is
a) differentiable at x = 0 and x = 1
b) differentiable at x = 0 not differentiable at x = 2
c) differentiable at x = 2 and not differentiable at x = 1
d) differentiable neither at x = 0 nor at x = 2.

Explanation:

2. If $f\left(x\right)=\cot^{-1}\left(\frac{x^{x}-x^{-x}}{2}\right)$     then f' (1) equals
a) -1
b) 1
c) log 2
d) – log 2.

Explanation: Putting u = x x and using logarithmic

3. If $f\left(x\right)=\left(1+x\right)^{n}$   , then the value of $f\left(0\right)+f'\left(0\right)+\frac{f"\left(0\right)}{2!}+....+\frac{\left(0\right)}{}$       is
a) n
b) $2^{n}$
c) $2^{n+1}$
d) none of these

Explanation:

4. The solution set of f '(x) > g'(x) where $f\left(x\right)=\left(1/2\right)5^{2x+1}$     and $g\left(x\right)=5^{x}+4x\log 5$     is
a) $\left(1,\infty\right)$
b) $\left(0,1\right)$
c) $\left[0,\infty\right)$
d) $\left(0,\infty\right)$

Explanation:

5.Let $f\left(x\right)=\sin x;g\left(x\right)=x^{2}$     and h(x) = log x. If u (x) = h( f (g(x))), then $\frac{d^{2}u}{dx^{2}}\mid x=\sqrt{\pi}/2$     is
a) $2-\pi$
b) $2-2\pi$
c) $\pi-2$
d) $2\pi-2$

Explanation:

6. If $y=\tan^{-1}\frac{1}{1+x+x^{2}}+\tan^{-1}\frac{1}{x^{2}+3x+3}+\tan^{-1}\frac{1}{x^{2}+5x+7}+....+$
up to n terms, then y'(0) is equal to
a) $-\frac{1}{1+n^{2}}$
b) $-\frac{n^{2}}{1+n^{2}}$
c) $\frac{n}{1+n^{2}}$
d) none of these

Explanation:

7. Let f and g be functions satisfying
$f\left(x\right)=e^{x}g\left(x\right),f\left(x+y\right)=f\left(x\right)+f\left(y\right),g\left(0\right)=0 , g'\left(0\right)=4$
g and g' are continuous at 0. Then
a) f (x) = 0 for all x
b) f (x) = x for all x
c) f (x) = x + 4 for all x
d) f (x) = 4 x for all x

Explanation:

8. Let $f:R\rightarrow R$   is a function which is defined by $f\left(x\right)=\max\left\{x,x^{3}\right\}$    . The set of all points on which f (x) is not differentiable is
a) $\left\{-1,1\right\}$
b) $\left\{-1,0\right\}$
c) $\left\{0,1\right\}$
d) $\left\{-1,0\right\}$

Explanation: The graphs of y = x and y = x3 are given in the

9. Which of the following functions is differentiable at x = 0?
a) cos (|x|) + |x|
b) cos (|x|) – |x|
c) sin |x| + |x|
d) sin (|x|) – |x|

10. If f is a differentiable function at a point 'a’ and $f'\left(a\right)\neq0$   then which of the following is true.
a) $-f'\left(a\right)=\lim_{h \rightarrow 0}\frac{f\left(a\right)-f\left(a-h\right)}{h}$
b) $\frac{1}{2}f'\left(a\right)=\lim_{h \rightarrow 0}\frac{f\left(a+2h\right)-f\left(a+h\right)}{2h}$
c) $f'\left(a\right)=\lim_{h \rightarrow 0}\frac{f\left(a+2h\right)-f\left(a\right)}{h}$
d) $3f'\left(a\right)=\lim_{h \rightarrow \infty}\frac{f\left(a+3h\right)-f\left(a+h\right)}{h}$