Applications Of Derivatives Question 191
Question: If $ f(x+y)=f(x).f(y) $ for all x and y and $ f(5)=2 $ , $ f’(0)=3 $ , then $ f’(5) $ will be
[IIT 1981; Karnataka CET 2000; UPSEAT 2002; MP PET 2002; AIEEE 2002]
Options:
A) 2
B) 4
C) 6
D) 8
Show Answer
Answer:
Correct Answer: C
Solution:
Let $ x=5,,y=0\Rightarrow f(5+0)=f(5).f(0) $
therefore $ f(5)=f(5)f(0)\Rightarrow f(0)=1 $
Therefore, $ f’(5)=\underset{h\to 0}{\mathop{\lim }},\frac{f(5+h)-f(5)}{h} $
$ =\underset{h\to 0}{\mathop{\lim }},\frac{f(5)f(h)-f(5)}{h}=\underset{h\to 0}{\mathop{\lim 2}},[ \frac{f(h)-1}{h} ] $ , $ { \because f(5)=2 } $
$ =2\underset{h\to 0}{\mathop{\lim }},.[ \frac{f(h)-f(0)}{h} ]=2\times f’(0)=2\times 3=6 $ .
BETA
