The differential coefficient of log₁₀ x with respect to log_x10 is

(a)

1

(b)

-(log₁₀ x)²

(c)

(log_x 10)²

(d)

\(x^2\over100\)

If f(x) = x + 2, then f '(f(x)) at x = 4 is

(a)

8

(b)

1

(c)

4

(d)

5

It is given that f '(a) exists, then \(lim_{x\rightarrow a}{xf(a)-af(x)\over x-a}\) is

(a)

f(a) - af '(a)

(b)

f '(a)

(c)

- f '(a)

(d)

f(a) + af '(a)

If \(f(x)=\left\{\begin{array}{l} x+1, \quad \text { when } x<2 \\ 2 x-1 \text { when } x \geq 2 \end{array}\right.\), then f'(2) is

(a)

0

(b)

1

(c)

2

(d)

does not exist

If g(x) = (x² + 2x + 3) f(x) and f(0) = 5 and \(lim_{x \rightarrow 0}{f(x)-5\over x}=4\), then g'(0) is

(a)

20

(b)

14

(c)

18

(d)

12

If f(x) = \(\left\{\begin{matrix} x+2& -1<x<3\\ 5,& x=3\\ 8-x,& x>3\\ \end{matrix}\right.\), then at x = 3, f'(x) is:

(a)

1

(b)

-1

(c)

0

(d)

does not exist

The derivative of f(x) = x |x| at x = −3 is

(a)

6

(b)

-6

(c)

does not exist

(d)

0

If \(f(x)= \begin{cases}2 a-x, & \text { for } \quad-a<x<a \\ 3 x-2 a & \text { for } \quad x \geq a\end{cases}\), then which one of the following is true?

(a)

f(x) is not differentiable at x = a

(b)

f(x) is discontinuous at x = a

(c)

f(x) is continuous for all x in R

(d)

f(x) is differentiable for all x \(\ge\) a

\(\text { If } f(x)=\left\{\begin{array}{ll} a x^2-b, & -1<x<1 \\ \frac{1}{|x|}, & \text { elsewhere } \end{array} \ \text { is differentiable at } x=1\right. \text {, then }\) 

(a)

\(a={1\over2},b={-3\over 2}\)

(b)

\(a={-1\over2},b={3\over 2}\)

(c)

\(a=-{1\over2},b=-{3\over 2}\)

(d)

\(a={1\over2},b={3\over 2}\)

The number of points in R in which the function \(f(x)=|x-1|+|x-3|+sin \ x\) is not differentiable, is

(a)

3

(b)

2

(c)

1

(d)

4