Let fx be a function satisfying fx-y-fxfy for all x-y -R and fx-1-xgx- where limx- 0gx-1- then f-x is equal to

The correct option is B f(x) #10;f'(x)=lim_h → #160;0f (x+h) f(x)/h #10; #10; #160;=lim_h→ 0f(x) #10;f(h) f(x)/h #10; #10; # ...

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Byju's Answer

Standard XII

Mathematics

Continuity of a Function

Let fx be a...

Question

Let $f (x)$ be a function satisfying $f (x + y) = f (x) f (y)$ for all $x, y$ $\in R$ and $f (x) = 1 + x g (x)$ , where $lim x \to 0 g (x) = 1$ , then $f^{'} (x)$ is equal to

x g (x)

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g^{'} (x)

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f (x)

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0

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Solution

The correct option is B $f (x)$
$f^{'} (x) = lim h \to 0 \frac{f (x + h) - f (x)}{h}$

$= lim h \to 0 \frac{f (x) f (h) - f (x)}{h}$

$= f (x) lim h \to 0 \frac{f (h) - 1}{h}$

$= f (x) lim h \to 0 g (h)$ , $(∵ f (x) = 1 + x g (x))$

$= f (x)$ ( $∵ lim x \to 0 g (x) = 1$ )

$\Rightarrow f^{'} (x) = f (x)$

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Similar questions

Q. Let

f (x + y) = f (x) f (y) \forall x, y ϵ R

and

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

where

l i m x \to 0 g (x) = 1

. Then:

Q. Let

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

is equal to

Q. Let

f (x)

be continuous and differentiable function satisfying

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

for all

x, y ϵ R

.
If

f (x)

can be expressed as

f (x) = 1 + x p (x) + x^{2} q (x)

where

lim x \to 0 p (x) = a

and

lim x \to 0 q (x) = b

then

f^{'} (x)

Q. If

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

for all

x, y ϵ R

and

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

, where

lim x \to 0 g (x) = 0

and

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exists, prove that

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Q. Let

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

and

f (x) = x^{2} g (x)

for all

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, where

g (x)

is continuous function. Then

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Let f(x) be a function satisfying f(x+y)=f(x)f(y) for all x,y ∈R and f(x)=1+xg(x), where limx→0g(x)=1, then f′(x) is equal to

The correct option is B f(x)f′(x)=limh→0f(x+h)−f(x)h =limh→0f(x)f(h)−f(x)h =f(x)limh→0f(h)−1h =f(x)limh→0g(h), (∵f(x)=1+xg(x)) =f(x) (∵limx→0g(x)=1) ⇒f′(x)=f(x)

Let $f (x)$ be a function satisfying $f (x + y) = f (x) f (y)$ for all $x, y$ $\in R$ and $f (x) = 1 + x g (x)$ , where $lim x \to 0 g (x) = 1$ , then $f^{'} (x)$ is equal to