The following function on the domain t>0 has a local maximum at t=1/2. True of False?

$\displaystyle

f(t) = \sqrt t .e^{ - t}

$

When I differentiate I get the following, which I think is correct.

$\displaystyle

f'(t) = e^{ - t} (\frac{1}{2}t^{\frac{{ - 1}}{2}} - t^{\frac{-1}{2}} )

$

However I cannot see how t=1/2 is a critical point. I know critical point occur when 1st derivative is equal to zero however, I doesn't appear to me that t=1/2 makes f'(t) zero