Prove that $(P-t)\Phi(\frac{t-P}{\sqrt{T(t)}})+\frac{1}{\sqrt{T(t)}}\phi(\frac{t-P}{\sqrt{T(t)}}) <0$ if $t$ is large enough

Question

I need to show

$f(t)=(P-t)\Phi(\frac{t-P}{\sqrt{T(t)}})+\frac{1}{\sqrt{T(t)}}\phi(\frac{t-P}{\sqrt{T(t)}}) <0$ if $t$ is large enough

where $T(t)>0$ and is a function of $t$, with $t>P>0$, and $\Phi, \phi$ are the normal cdf and pdf respectively.

I am totally stumped.

I know that when $t=P$, $f(P)=\frac{1}{\sqrt{2\pi T(P)}}>0$ but I can't think of anything else due to the implicit nature of the problem.