Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 5 - Continuity and Differentiability>Target Exercises>Q 59

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If $f (x + y) = 2 f (x) \cdot f (y)$ for all $x, y$ , where $f^{'} (0) = 3$ and $f (4) = 2$ then $f^{'} (4)$ is equal to

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Important Questions on Continuity and Differentiability

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JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 5 - Continuity and Differentiability>Target Exercises>Q 60

If a function

f : R \to R

be such that

f (x + y) = f (x) \cdot f (y)

for all

x, y \in R

, where

f (x) = 1 + x ϕ (x)

and

\lim_{x \to 0} ϕ (x) = 1

, then

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JEE Advanced

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Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 5 - Continuity and Differentiability>Target Exercises>Q 61

Let

f (x)

be a function satisfying

f (x + y) = f (x) f (y)

for all

x, y \in R

. If

f (x) = 1 + x ϕ (x) + x^{2} ϕ (x) ψ (x)

, where

\lim_{x \to 0} ϕ (x) = a

and

\lim_{x \to 0} ψ (x) = b

, then

f^{'} (x)

is equal to

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Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 5 - Continuity and Differentiability>Target Exercises>Q 62

Let

f

be a function satisfying

f (x + y) = f (x) + f (y)

and

f (x) = x^{3} ϕ (x)

for all

x

and

y

, where

ϕ (x)

is a continuous function, then

f^{'} (x)

is equal to

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JEE Advanced

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Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 5 - Continuity and Differentiability>Target Exercises>Q 63

Let

f (x + y) = f (x) \cdot f (y)

for all

x, y \in R

and

f (x) = 1 + x ϕ (x) \log 2

, where

\lim_{x \to 0} ϕ (x) = 1

. Then

f^{'} (x)

is equal to

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JEE Advanced

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Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 5 - Continuity and Differentiability>Target Exercises>Q 64

Let

f (x)

be a real function not identically zero in

Z

, such that

f (x + y^{2 n + 1}) = f (x) + {\{f (y)\}}^{2 n + 1}

n \in N

and

x, y \in R

. If

f^{'} (0) \geq 0

, then

f^{'} (6)

is equal to

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JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 5 - Continuity and Differentiability>Target Exercises>Q 65

Let

f (x + y) = f (x) \cdot f (y)

and

f (x) = 1 + x g (x) G (x)

, where

\lim_{x \to 0} g (x) = a

and

\lim_{x \to 0} G (x) = b

. Then

f^{'} (x) = k f (x)

, where

k

is equal to

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JEE Advanced

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Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 5 - Continuity and Differentiability>Target Exercises>Q 66

If

f (x) = [\log_{e} x] + \sqrt{\{\log_{e} x\}}, x > 1

, where

[]

and

\{\}

denote the greatest integer function and the fractional part function respectively, then

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Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 5 - Continuity and Differentiability>Target Exercises>Q 67

If

f (2 + x) = f (- x)

for all

x \in R

, then differentiability at

x = 4

implies differentiability at