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Continuity and Differentiability

Multiple Choice Questions

31.

$Let f : ℝ \to ℝ be a function such that f (x) = x^{3} + x^{2} f' (1) + xf'' (2) + f''' (3) for x \in ℝ$

Consider the following :

f(2) = f(1) - f(0)
f"(2) - 2f'(1) = 12

Which of the above is/are correct ?

1 only
2 only
Both 1 and 2
Neither 1 nor 2

32.

Consider the following in respect of the function

$f (x) = \{2 + x, x \geq 0 2 - x, x < 0\} 1 . \lim_{x \to 1} f (x) does not exist 2 . f (x) is differentiable at x = 0 3 . f (x) is continuous at x = 0$

Which of the above statements is/are correct ?

1 only
3 only
2 and 3 only
1 and 3 only

33.

Let $f : A \to ℝ, where A = ℝ ╲ (0) is such that f (x) = \frac{x + |x|}{x} .$

On which one of the following sets is f (x) continuous ?

A
$B = \{x \in ℝ : x \geq 0\}$
$C = \{x \in ℝ : x \leq 0\}$
$D = ℝ$

34.

$f (x) = \{3 x^{2} + 12 x - 1, - 1 \leq x \leq 2 37 - x, 2 < x \leq 3\}$

Which of the following statements is/are correct ?

f(x) is continuous at x = 2.
f(x) attains greatest value at x = 2.
f(x) is differentiable at x = 2

Select the correct answer using the code given below :

1 and 3 only
2 and 3 only
Both 1 and 2
1, 2 and 3

35.

Which of the following equations is/are correct ?

1. f( - 2) = f(5)

2. f"( - 2) + f"(0.5) + f"(3) = 4

Select the correct answer using the code given below :

1 Only
2 Only
Both 1 and 2
Neither 1 nor 2

36.
Let f(x) = [x], where [.] is the greatest integer function and g(x) = sin(x) be two real valued functions over $ℝ$
Which of the following statements is correct ?

Both f(x) and g(x) are continuous at x = 0.

f(x) is continuous at x = 0, but g(x) is not continuous at x = 0

g(x) is continuous at x = 0, but f(x) is not continuous at x = 0

Both f(x) and g(x) are discontinuous at x = 0

g(x) is continuous at x = 0, but f(x) is not continuous at x = 0

$f (x) = |x| and g (x) = \sin (x) \lim_{x \to 0^{+}} f (x) = |0 + h| = 0 \lim_{x \to 0^{-}} f (x) = |0 - h| = - 1 f (0) - 0 \Rightarrow f (x) is not continuous at x = 0 and also g (x) is continuous at x = 0 . (every trignometric function is continuous) .$