Applications Of Derivatives Question 261
Question: If $ f(x)=x{e^{x(1-x)}} $ , then $ f(x) $ is
[IIT Screening 2001]
Options:
A) Increasing on $ [ -\frac{1}{2},,1 ] $
B) Decreasing on R
C) Increasing on R
D) Decreasing on $ [ -\frac{1}{2},1 ] $
Show Answer
Answer:
Correct Answer: A
Solution:
$ {f}’(x)={e^{x(1-x)}}+x.{e^{x(1-x)}}.(1-2x) $
$ ={e^{x(1-x)}}{1+x(1-2x)}={e^{x(1-x)}}.(-2x^{2}+x+1) $
Now by the sign-scheme for $ -2x^{2}+x+1 $
$ {f}’(x)\ge 0, $ if $ x,\in ,[ -\frac{1}{2},,1 ], $ because $ e^{x}(1-x) $ is always positive. So, $ f(x) $ is increasing on $ [ -\frac{1}{2},,1 ] $ .
BETA
