How do you find all intervals where the function #f(x)=e^(x^2)# is increasing?

1 Answer
Jim H
Aug 26, 2015

Investigate the sign of #f'(x)#.

Explanation:

On intervals on which #f'(x)# is positive (#>0#), #f(x)# is increasing.

#f(x)=e^(x^2)#

#f'(x)=2xe^(x^2)#

Because #f'(x)# is never undefined, it could possibly change sign only at #x# values where #f'(x) =0#

#2xe^(x^2) = 0# if and only if

#2x = 0#, so #x = 0# #

or #e^(x^2) = 0# but #e^n# is never #0# for any #n#.

Since #2e^(x^2) > 0# for all #x#, the sign of #f'(x)# is the same as the sign of #x#

which is (of course) negative for #x<0# and positive for #x>0#.

So #f# is increasing on the interval #(0, oo)#.

(In my experience the usual practice is to state open intervals on which a function is increasing. It is also true that this function is increasing on the closed interval: #[0,oo)#.)

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