Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 2>Q 5

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Suppose $f$ is a real valued function and $c$ is a point in its domain. The derivative of $f$ at $c$ (if limit exist) is defined by

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Important Questions on Continuity and Differentiability

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Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 2>Q 6

If both

\lim_{h \to 0} \frac{f (c - h) - f (c)}{- h}

and

\lim_{h \to 0} \frac{f (c + h) - f (c)}{h}

are finite and equal, then

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Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 2>Q 7

If the function

f (x) = \{\begin{matrix} 1 & , & x \leq 2 \\ a x + b & , & 2 < x < 4 \\ 7 & , & x \geq 4 \end{matrix}

is continuous at

x = 2

and

4

, then the values of

a

and

b

are

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Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 2>Q 8

If

f (x) = \{\begin{matrix} \frac{[x] - 1}{x - 1}, x \neq 1 \\ 0, x = 1 \end{matrix}

where

[\cdot]

denotes the greatest integer function, then

f (x)

is

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Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 2>Q 9

If

f (x) = \{\begin{cases} x^{k} \cos (\frac{1}{x}), & x \neq 0 \\ 0 & x = 0 \end{cases}

is continuous at

x = 0

, then

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Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 2>Q 10

If

f (x) = \frac{1}{1 - x},

then the points of discontinuity of the function

f [f \{f (x)\}]

are

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Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 2>Q 11

Let

f (x)

be a even differentiable function. Consider the following statements:
(I)

f^{'} (x)

is an even function
(II)

f' (x)

is an odd function
(III)

f^{'} (x)

may be even or odd. Which of the above statements is/are correct?

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Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 2>Q 12

Consider the following statements:
I. The function $f (x) =$ greatest integer $\leq x, x \in R$ is a continuous function.

II. All trigonometric functions are continuous on $R$ .

Which of the statements given above is/are correct?

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Comprehensive Guide to AP EAPCET Mathematics>Chapter 9 - Continuity and Differentiability>Exercise 2>Q 13

If $f (x) = \{\begin{cases} x + λ, & x < 3 \\ 4, & x = 3 \\ 3 x - 5, & x > 3 \end{cases}$ is continuous at $x = 3,$ then the value of $λ$ is equal to