To determine the values of
where
is increasing, we must solve for the critical points of
and examine the sign of the derivative in between the critical points, solving
gives
![{\displaystyle {\begin{aligned}\displaystyle f'(t)&=0\\2t\exp(t)+t^{2}\exp(t)&=0\\(t^{2}+2t)\exp(t)&=0.\end{aligned}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/5cf9bd072d3b31b5663309739b3f5b70a2d85789)
Since the exponential function is never zero for any value of
, we can solve the above equation by solving
![{\displaystyle {\begin{aligned}\displaystyle t^{2}+2t&=0\\t(t+2)&=0,\quad \rightarrow \quad t=0,\,-2\end{aligned}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/12780ec597135f811b187a236d563ac257b529ac)
Now we need to evaluate the sign of
in the intervals
. Taking test points
and plugging them into
gives:
![{\displaystyle \displaystyle f'(-3)=3e^{-3}>0,\quad f'(-1)=-e^{-1}<0,\quad f'(1)=3e>0.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/a2d6414d6b8617965d6c882e87a6adeb0577864c)
So
is positive on
and negative on
.
Therefore,
is increasing for
,
and decreasing for
.