increasing function

[Var1995]

can someone pls do it here? i m going somewhere wrong while calculating the derivative(/ the fn) but m not able to detect it. someone pls solve it here so that i can check my mistake(s).

[Akash]

[Dr. Strange]

Answer should be x<0
condition for increasing will be 2*x [ exp -(x^2+1)^2 - exp -(x^4) ] >0
Case 1 : when x>0
Then  exp -(x^2+1)^2 - exp -(x^4)  >0
exp ( x^4 - (x^2+1)^2 )  -1 >0
So exp (x^4 - (x^2+1)^2)>1
so x^4 - (x^2+1)^2>0
-(2 x^2 +1) >0
which is not possible

Case 2:  when x<0
exp -(x^2+1)^2 - exp -(x^4)  <0
exp ( x^4 - (x^2+1)^2 )  -1 <0
So exp (x^4 - (x^2+1)^2)<1
so x^4 - (x^2+1)^2<0
-(2 x^2 +1) <0

which is always true
so function is increasing when x<0

[Akash]

Yeah you are correct. I forgot the 2x term