## Limits, Continuity and Differentiability Questions and Answers Part-15

1. If f'(x) = g(x) and g'(x) = - f (x) for all x and f (2) = 4 = f' (2) then $f^{2}\left(8\right)+g^{2}\left(8\right)$    is
a) 16
b) 32
c) 64
d) 8

Explanation:

2. Let $f\left(x\right)=\begin{cases}\sin^{2}x\left(\frac{e^{3/x}-e^{-3/x}}{e^{3/x}+e^{-3/x}}\right) & x \neq 0\\0 & x = 0\end{cases}$
then
a) f is not continuous at x = 0
b) f is continuous but not differentiable at x = 0
c) f is differentiable at x = 0
d) f is differentiable at x = 0 and f'(0) = 3

Explanation:

3. If $x=2\sin t-\sin 2t,y=2\cos t-\cos2t,$
then the value of $\frac{d^{2}y}{dx^{2}}$   at $t=\pi$  is
a) 2
b) 1/8
c) -3/4
d) -3/2

Explanation:

4. $f\left(x\right)=\begin{cases}\mid x-4\mid & x \geq 1\\\left(x^{3}/2\right)-x^{2}+3x+1/2 & x < 1\end{cases}$
then
a) f (x) is continuous at x = 1 and x = 4
b) f (x) is differentiable at x = 4
c) f (x) is continuous and differentiable at x = 1
d) f (x) is only continuous at x = 1

Explanation:

5. If $f\left(x\right)=x^{2}+\frac{x^{2}}{\left(1+x^{2}\right)}+\frac{x^{2}}{\left(1+x^{2}\right)^{2}}+....+\frac{x^{2}}{\left(1+x^{2}\right)^{n}}+....$
then at x =0
a) f (x) has no limit
b) f (x) is discontinuous
c) f (x) is continuous but not differentiable
d) (x) is differentiable

Explanation:

6. Let $f\left(x\right)=\begin{cases}\int_{0}^{x}\left\{1+\mid 1-t\mid\right\}dt & x > 2\\5x-7 & x \leq 2\end{cases}$
then
a) f is not continuous at x = 2
b) f is continuous but not differentiable at x = 2
c) f is differentiable everywhere
d) f'(2+) doesn’t exist

Explanation:

7. If $f\left(x\right)=\mid \log_{5}\left(x^{3}+10x^{2}+11x-69\right)\mid$
then at x = 2
a) f is not continuous
b) f is continuous but not differentiable
c) f is differentiable
d) the derivative is 1

Explanation:

8. If f''(x) is continuous at x = 0 and f ''(0) = 5, then the value of $\lim_{x \rightarrow 0}\frac{2f\left(x\right)-3f\left(2x\right)+f\left(4x\right)}{x^{2}}$
is
a) 10
b) 8
c) 15
d) 12

Explanation:

9. 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) $\left(0,\infty\right)$

10. Suppose f is differentiable at x = 2 and $\lim_{h \rightarrow0}\frac{f\left(2+h\right)}{h}=3$