Applications Of Derivatives Question 242
Question: The function f defined by $ f(x)=(x+2){e^{-x}} $ is
[IIT Screening 1994]
Options:
A) Decreasing for all x
B) Decreasing in $ (-\infty ,,-1) $ and increasing in $ (-1,\infty ) $
C) Increasing for all x
D) Decreasing in $ (-1,,\infty ) $ and increasing in $ (-\infty ,,-1) $
Show Answer
Answer:
Correct Answer: D
Solution:
$ f(x)=(x+2){e^{-x}} $
$ f’(x)={e^{-x}}-{e^{-x}}(x+2) $
$ f’(x)=-{e^{-x}}[x+1] $
For increasing, $ -{e^{-x}}(x+1)>0 $ or $ {e^{-x}}(x+1)<0 $
$ {e^{-x}}>0 $
$ (x+1)<0 $
$ x\in (-\infty ,,\infty ) $ and $ x\in (-\infty ,-1) $
$ \therefore x\in (-\infty ,-1) $
Hence, the function is increasing in $ (-\infty ,,-1) $
For decreasing, $ -{e^{-x}}(x+1)<0 $ or $ {e^{-x}}(x+1)>0 $ , $ x\in (-1,,\infty ) $
Hence the function is decreasing in $ (-1,\ \infty ) $ .
BETA
